AMOEBA uses fast approximation for erfc()

plugins/amoeba/platforms/cuda/src/kernels/multipoleFixedField.cu

@@ -40,10 +40,20 @@ __device__ void computeOneInteraction(AtomData& atom1, AtomData& atom2, real3 de
 
         real r = SQRT(r2);
         real ralpha = EWALD_ALPHA*r;
-        real bn0 = erfc(ralpha)/r;
+        real exp2a = EXP(-(ralpha*ralpha));
+#ifdef USE_DOUBLE_PRECISION
+        const real erfcAlphaR = erfc(ralpha);
+#else
+        // This approximation for erfc is from Abramowitz and Stegun (1964) p. 299.  They cite the following as
+        // the original source: C. Hastings, Jr., Approximations for Digital Computers (1955).  It has a maximum
+        // error of 1.5e-7.
+
+        const real t = RECIP(1.0f+0.3275911f*ralpha);
+        const real erfcAlphaR = (0.254829592f+(-0.284496736f+(1.421413741f+(-1.453152027f+1.061405429f*t)*t)*t)*t)*t*exp2a;
+#endif
+        real bn0 = erfcAlphaR/r;
         real alsq2 = 2*EWALD_ALPHA*EWALD_ALPHA;
         real alsq2n = RECIP(SQRT_PI*EWALD_ALPHA);
-        real exp2a = EXP(-(ralpha*ralpha));
         alsq2n *= alsq2;
         real bn1 = (bn0+alsq2n*exp2a)/r2;
         alsq2n *= alsq2;

plugins/amoeba/platforms/cuda/src/kernels/multipoleInducedField.cu

@@ -62,10 +62,20 @@ __device__ void computeOneInteraction(AtomData& atom1, AtomData& atom2, real3 de
         // calculate the error function damping terms
 
         real ralpha = EWALD_ALPHA*r;
-        real bn0 = erfc(ralpha)*rI;
+        real exp2a = EXP(-(ralpha*ralpha));
+#ifdef USE_DOUBLE_PRECISION
+        const real erfcAlphaR = erfc(ralpha);
+#else
+        // This approximation for erfc is from Abramowitz and Stegun (1964) p. 299.  They cite the following as
+        // the original source: C. Hastings, Jr., Approximations for Digital Computers (1955).  It has a maximum
+        // error of 1.5e-7.
+
+        const real t = RECIP(1.0f+0.3275911f*ralpha);
+        const real erfcAlphaR = (0.254829592f+(-0.284496736f+(1.421413741f+(-1.453152027f+1.061405429f*t)*t)*t)*t)*t*exp2a;
+#endif
+        real bn0 = erfcAlphaR*rI;
         real alsq2 = 2*EWALD_ALPHA*EWALD_ALPHA;
         real alsq2n = RECIP(SQRT_PI*EWALD_ALPHA);
-        real exp2a = EXP(-(ralpha*ralpha));
         alsq2n *= alsq2;
         real bn1 = (bn0+alsq2n*exp2a)*rI*rI;
 

plugins/amoeba/platforms/cuda/src/kernels/pmeMultipoleElectrostatics.cu

@@ -92,7 +92,17 @@ __device__ void computeOneInteraction(AtomData& atom1, AtomData& atom2, bool has
 
     real rr1 = RECIP(r);
     delta.w = rr1;
-    real bn0 = erfc(ralpha)*rr1;
+#ifdef USE_DOUBLE_PRECISION
+    const real erfcAlphaR = erfc(ralpha);
+#else
+    // This approximation for erfc is from Abramowitz and Stegun (1964) p. 299.  They cite the following as
+    // the original source: C. Hastings, Jr., Approximations for Digital Computers (1955).  It has a maximum
+    // error of 1.5e-7.
+
+    const real t = RECIP(1.0f+0.3275911f*ralpha);
+    const real erfcAlphaR = (0.254829592f+(-0.284496736f+(1.421413741f+(-1.453152027f+1.061405429f*t)*t)*t)*t)*t*exp2a;
+#endif
+    real bn0 = erfcAlphaR*rr1;
     energy += forceFactor*atom1.q*atom2.q*bn0;
     real rr2 = rr1*rr1;
     alsq2n *= alsq2;