Average Error: 24.8 → 8.3
Time: 8.5s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\ \;\;\;\;x - \mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\

\mathbf{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\
\;\;\;\;x - \mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r285509 = x;
        double r285510 = 1.0;
        double r285511 = y;
        double r285512 = r285510 - r285511;
        double r285513 = z;
        double r285514 = exp(r285513);
        double r285515 = r285511 * r285514;
        double r285516 = r285512 + r285515;
        double r285517 = log(r285516);
        double r285518 = t;
        double r285519 = r285517 / r285518;
        double r285520 = r285509 - r285519;
        return r285520;
}
↓
double f(double x, double y, double z, double t) {
        double r285521 = z;
        double r285522 = -5.050049854843211e-06;
        bool r285523 = r285521 <= r285522;
        double r285524 = x;
        double r285525 = 1.0;
        double r285526 = y;
        double r285527 = r285525 - r285526;
        double r285528 = exp(r285521);
        double r285529 = r285526 * r285528;
        double r285530 = cbrt(r285529);
        double r285531 = r285530 * r285530;
        double r285532 = r285531 * r285530;
        double r285533 = r285527 + r285532;
        double r285534 = log(r285533);
        double r285535 = t;
        double r285536 = r285534 / r285535;
        double r285537 = r285524 - r285536;
        double r285538 = 1.8264381599139905e-113;
        bool r285539 = r285521 <= r285538;
        double r285540 = 0.5;
        double r285541 = 2.0;
        double r285542 = pow(r285521, r285541);
        double r285543 = r285542 * r285526;
        double r285544 = r285521 * r285526;
        double r285545 = log(r285525);
        double r285546 = fma(r285525, r285544, r285545);
        double r285547 = fma(r285540, r285543, r285546);
        double r285548 = 1.0;
        double r285549 = r285548 / r285535;
        double r285550 = r285547 * r285549;
        double r285551 = r285524 - r285550;
        double r285552 = 0.5;
        double r285553 = fma(r285521, r285526, r285525);
        double r285554 = fma(r285552, r285543, r285553);
        double r285555 = log(r285554);
        double r285556 = r285555 / r285535;
        double r285557 = r285524 - r285556;
        double r285558 = r285539 ? r285551 : r285557;
        double r285559 = r285523 ? r285537 : r285558;
        return r285559;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original24.8
Target15.8
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.050049854843211e-06

    1. Initial program 11.9

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt11.9

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}}\right)}{t}\]

    if -5.050049854843211e-06 < z < 1.8264381599139905e-113

    1. Initial program 30.6

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 5.8

      \[\leadsto x - \frac{\color{blue}{0.5 \cdot \left({z}^{2} \cdot y\right) + \left(1 \cdot \left(z \cdot y\right) + \log 1\right)}}{t}\]

    3. Simplified5.8

      \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}}{t}\]
    4. Using strategy rm

    5. Applied div-inv5.8

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}}\]

    if 1.8264381599139905e-113 < z

    1. Initial program 28.7

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 11.8

      \[\leadsto x - \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + \left(z \cdot y + 1\right)\right)}}{t}\]

    3. Simplified11.8

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\ \;\;\;\;x - \mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t)))

  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))