Average Error: 13.8 → 13.9
Time: 23.4s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[1 - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}}\right) \cdot 1.061405428999999900341322245367337018251 + -1.453152027000000012790792425221297889948\right) + 1.421413741000000063863240029604639858007\right)\right) + \mathsf{fma}\left(-0.2844967359999999723108032867457950487733, \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 0.2548295919999999936678136691625695675611\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right)\]
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|}
↓
1 - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(\left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}}\right) \cdot 1.061405428999999900341322245367337018251 + -1.453152027000000012790792425221297889948\right) + 1.421413741000000063863240029604639858007\right)\right) + \mathsf{fma}\left(-0.2844967359999999723108032867457950487733, \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 0.2548295919999999936678136691625695675611\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right)
double f(double x) {
        double r133113 = 1.0;
        double r133114 = 0.3275911;
        double r133115 = x;
        double r133116 = fabs(r133115);
        double r133117 = r133114 * r133116;
        double r133118 = r133113 + r133117;
        double r133119 = r133113 / r133118;
        double r133120 = 0.254829592;
        double r133121 = -0.284496736;
        double r133122 = 1.421413741;
        double r133123 = -1.453152027;
        double r133124 = 1.061405429;
        double r133125 = r133119 * r133124;
        double r133126 = r133123 + r133125;
        double r133127 = r133119 * r133126;
        double r133128 = r133122 + r133127;
        double r133129 = r133119 * r133128;
        double r133130 = r133121 + r133129;
        double r133131 = r133119 * r133130;
        double r133132 = r133120 + r133131;
        double r133133 = r133119 * r133132;
        double r133134 = r133116 * r133116;
        double r133135 = -r133134;
        double r133136 = exp(r133135);
        double r133137 = r133133 * r133136;
        double r133138 = r133113 - r133137;
        return r133138;
}
↓
double f(double x) {
        double r133139 = 1.0;
        double r133140 = x;
        double r133141 = fabs(r133140);
        double r133142 = -r133141;
        double r133143 = r133142 * r133141;
        double r133144 = exp(r133143);
        double r133145 = 0.3275911;
        double r133146 = r133141 * r133145;
        double r133147 = r133139 + r133146;
        double r133148 = r133139 / r133147;
        double r133149 = sqrt(r133139);
        double r133150 = fma(r133141, r133145, r133139);
        double r133151 = cbrt(r133150);
        double r133152 = r133149 / r133151;
        double r133153 = r133151 * r133151;
        double r133154 = r133149 / r133153;
        double r133155 = r133152 * r133154;
        double r133156 = 1.061405429;
        double r133157 = r133155 * r133156;
        double r133158 = -1.453152027;
        double r133159 = r133157 + r133158;
        double r133160 = r133148 * r133159;
        double r133161 = 1.421413741;
        double r133162 = r133160 + r133161;
        double r133163 = r133148 * r133162;
        double r133164 = r133148 * r133163;
        double r133165 = -0.284496736;
        double r133166 = r133139 / r133150;
        double r133167 = 0.254829592;
        double r133168 = fma(r133165, r133166, r133167);
        double r133169 = r133164 + r133168;
        double r133170 = r133169 * r133148;
        double r133171 = r133144 * r133170;
        double r133172 = r133139 - r133171;
        return r133172;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.8

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

  3. Applied distribute-rgt-in13.8

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

  4. Applied associate-+r+13.8

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

  5. Simplified13.8

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

  7. Applied add-cube-cbrt13.9

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

  8. Applied add-sqr-sqrt13.9

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

  9. Applied times-frac13.9

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

  10. Simplified13.9

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

  11. Simplified13.9

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

  12. Final simplification13.9

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))