Average Error: 13.9 → 13.9
Time: 24.0s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\log \left(e^{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, -\frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right)}\right)\]
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|}
↓
\log \left(e^{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, -\frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right)}\right)
double f(double x) {
        double r52491 = 1.0;
        double r52492 = 0.3275911;
        double r52493 = x;
        double r52494 = fabs(r52493);
        double r52495 = r52492 * r52494;
        double r52496 = r52491 + r52495;
        double r52497 = r52491 / r52496;
        double r52498 = 0.254829592;
        double r52499 = -0.284496736;
        double r52500 = 1.421413741;
        double r52501 = -1.453152027;
        double r52502 = 1.061405429;
        double r52503 = r52497 * r52502;
        double r52504 = r52501 + r52503;
        double r52505 = r52497 * r52504;
        double r52506 = r52500 + r52505;
        double r52507 = r52497 * r52506;
        double r52508 = r52499 + r52507;
        double r52509 = r52497 * r52508;
        double r52510 = r52498 + r52509;
        double r52511 = r52497 * r52510;
        double r52512 = r52494 * r52494;
        double r52513 = -r52512;
        double r52514 = exp(r52513);
        double r52515 = r52511 * r52514;
        double r52516 = r52491 - r52515;
        return r52516;
}
↓
double f(double x) {
        double r52517 = 1.0;
        double r52518 = x;
        double r52519 = fabs(r52518);
        double r52520 = 0.3275911;
        double r52521 = fma(r52519, r52520, r52517);
        double r52522 = r52517 / r52521;
        double r52523 = 1.061405429;
        double r52524 = -1.453152027;
        double r52525 = fma(r52522, r52523, r52524);
        double r52526 = 1.421413741;
        double r52527 = fma(r52522, r52525, r52526);
        double r52528 = 3.0;
        double r52529 = pow(r52522, r52528);
        double r52530 = cbrt(r52529);
        double r52531 = -0.284496736;
        double r52532 = fma(r52527, r52530, r52531);
        double r52533 = 0.254829592;
        double r52534 = fma(r52522, r52532, r52533);
        double r52535 = 2.0;
        double r52536 = pow(r52519, r52535);
        double r52537 = exp(r52536);
        double r52538 = r52534 / r52537;
        double r52539 = -r52538;
        double r52540 = fma(r52522, r52539, r52517);
        double r52541 = exp(r52540);
        double r52542 = log(r52541);
        return r52542;
}

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. Simplified13.9

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

  4. Applied add-log-exp13.9

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

  5. Simplified13.9

    \[\leadsto \log \color{blue}{\left(e^{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, -\frac{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{{\left(\left|x\right|\right)}^{2}}}, 1\right)}\right)}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube13.9

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

  8. Applied add-cbrt-cube13.9

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

  9. Applied cbrt-undiv13.9

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

  10. Simplified13.9

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

  11. Final simplification13.9

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

Reproduce

herbie shell --seed 2019315 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ 0.25482959199999999 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ -0.284496735999999972 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ 1.42141374100000006 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ -1.45315202700000001 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) 1.0614054289999999))))))))) (exp (- (* (fabs x) (fabs x)))))))