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 r236990 = x;
        double r236991 = 1.0;
        double r236992 = y;
        double r236993 = r236991 - r236992;
        double r236994 = z;
        double r236995 = exp(r236994);
        double r236996 = r236992 * r236995;
        double r236997 = r236993 + r236996;
        double r236998 = log(r236997);
        double r236999 = t;
        double r237000 = r236998 / r236999;
        double r237001 = r236990 - r237000;
        return r237001;
}
↓
double f(double x, double y, double z, double t) {
        double r237002 = z;
        double r237003 = -5.050049854843211e-06;
        bool r237004 = r237002 <= r237003;
        double r237005 = x;
        double r237006 = 1.0;
        double r237007 = y;
        double r237008 = r237006 - r237007;
        double r237009 = exp(r237002);
        double r237010 = r237007 * r237009;
        double r237011 = cbrt(r237010);
        double r237012 = r237011 * r237011;
        double r237013 = r237012 * r237011;
        double r237014 = r237008 + r237013;
        double r237015 = log(r237014);
        double r237016 = t;
        double r237017 = r237015 / r237016;
        double r237018 = r237005 - r237017;
        double r237019 = 1.8264381599139905e-113;
        bool r237020 = r237002 <= r237019;
        double r237021 = 0.5;
        double r237022 = 2.0;
        double r237023 = pow(r237002, r237022);
        double r237024 = r237023 * r237007;
        double r237025 = r237002 * r237007;
        double r237026 = log(r237006);
        double r237027 = fma(r237006, r237025, r237026);
        double r237028 = fma(r237021, r237024, r237027);
        double r237029 = 1.0;
        double r237030 = r237029 / r237016;
        double r237031 = r237028 * r237030;
        double r237032 = r237005 - r237031;
        double r237033 = 0.5;
        double r237034 = fma(r237002, r237007, r237006);
        double r237035 = fma(r237033, r237024, r237034);
        double r237036 = log(r237035);
        double r237037 = r237036 / r237016;
        double r237038 = r237005 - r237037;
        double r237039 = r237020 ? r237032 : r237038;
        double r237040 = r237004 ? r237018 : r237039;
        return r237040;
}

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)))