Jmat.Real.erf

Average Error: 13.9 → 13.9

Time: 11.2s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\frac{\mathsf{fma}\left(-{\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right)}^{3} \cdot {\left(\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}} \cdot \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), 1 \cdot 1\right)}\]

C

double f(double x) {
        double r270990 = 1.0;
        double r270991 = 0.3275911;
        double r270992 = x;
        double r270993 = fabs(r270992);
        double r270994 = r270991 * r270993;
        double r270995 = r270990 + r270994;
        double r270996 = r270990 / r270995;
        double r270997 = 0.254829592;
        double r270998 = -0.284496736;
        double r270999 = 1.421413741;
        double r271000 = -1.453152027;
        double r271001 = 1.061405429;
        double r271002 = r270996 * r271001;
        double r271003 = r271000 + r271002;
        double r271004 = r270996 * r271003;
        double r271005 = r270999 + r271004;
        double r271006 = r270996 * r271005;
        double r271007 = r270998 + r271006;
        double r271008 = r270996 * r271007;
        double r271009 = r270997 + r271008;
        double r271010 = r270996 * r271009;
        double r271011 = r270993 * r270993;
        double r271012 = -r271011;
        double r271013 = exp(r271012);
        double r271014 = r271010 * r271013;
        double r271015 = r270990 - r271014;
        return r271015;
}

↓

double f(double x) {
        double r271016 = 1.0;
        double r271017 = 0.3275911;
        double r271018 = x;
        double r271019 = fabs(r271018);
        double r271020 = r271017 * r271019;
        double r271021 = r271020 + r271016;
        double r271022 = r271016 / r271021;
        double r271023 = 3.0;
        double r271024 = pow(r271022, r271023);
        double r271025 = 1.061405429;
        double r271026 = -1.453152027;
        double r271027 = fma(r271022, r271025, r271026);
        double r271028 = 1.421413741;
        double r271029 = fma(r271022, r271027, r271028);
        double r271030 = -0.284496736;
        double r271031 = fma(r271022, r271029, r271030);
        double r271032 = 0.254829592;
        double r271033 = fma(r271022, r271031, r271032);
        double r271034 = pow(r271033, r271023);
        double r271035 = r271024 * r271034;
        double r271036 = -r271035;
        double r271037 = 1.0;
        double r271038 = 2.0;
        double r271039 = pow(r271019, r271038);
        double r271040 = exp(r271039);
        double r271041 = r271037 / r271040;
        double r271042 = pow(r271041, r271023);
        double r271043 = pow(r271016, r271023);
        double r271044 = fma(r271036, r271042, r271043);
        double r271045 = r271033 * r271016;
        double r271046 = fma(r271017, r271019, r271016);
        double r271047 = r271045 / r271046;
        double r271048 = -r271039;
        double r271049 = exp(r271048);
        double r271050 = r271016 + r271020;
        double r271051 = r271016 / r271050;
        double r271052 = r271041 * r271051;
        double r271053 = fma(r271052, r271033, r271016);
        double r271054 = r271049 * r271053;
        double r271055 = r271016 * r271016;
        double r271056 = fma(r271047, r271054, r271055);
        double r271057 = r271044 / r271056;
        return r271057;
}

Error

Bits error versus x

Derivation

1. Initial program 13.9

   \[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied flip3-- 13.9

   \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}}\]

4. Simplified 13.9

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-{\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right)}^{3} \cdot {\left(\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}}{1 \cdot 1 + \left(\left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}\]

5. Simplified 13.9

   \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right)}^{3} \cdot {\left(\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}} \cdot \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), 1 \cdot 1\right)}}\]

6. Final simplification 13.9

   \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right)}^{3} \cdot {\left(\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}, e^{-{\left(\left|x\right|\right)}^{2}} \cdot \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, \mathsf{fma}\left(\frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), 1 \cdot 1\right)}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))