Average Error: 25.3 → 8.6
Time: 25.1s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.184725816409017 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}, \sqrt[3]{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}, \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.184725816409017 \cdot 10^{-8}:\\
\;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}, \sqrt[3]{1 - y}, y \cdot e^{z}\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \mathsf{fma}\left(1, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}, \frac{\log 1}{t}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r166390 = x;
        double r166391 = 1.0;
        double r166392 = y;
        double r166393 = r166391 - r166392;
        double r166394 = z;
        double r166395 = exp(r166394);
        double r166396 = r166392 * r166395;
        double r166397 = r166393 + r166396;
        double r166398 = log(r166397);
        double r166399 = t;
        double r166400 = r166398 / r166399;
        double r166401 = r166390 - r166400;
        return r166401;
}
↓
double f(double x, double y, double z, double t) {
        double r166402 = z;
        double r166403 = -5.184725816409017e-08;
        bool r166404 = r166402 <= r166403;
        double r166405 = x;
        double r166406 = 1.0;
        double r166407 = y;
        double r166408 = r166406 - r166407;
        double r166409 = cbrt(r166408);
        double r166410 = r166409 * r166409;
        double r166411 = exp(r166402);
        double r166412 = r166407 * r166411;
        double r166413 = fma(r166410, r166409, r166412);
        double r166414 = log(r166413);
        double r166415 = t;
        double r166416 = r166414 / r166415;
        double r166417 = r166405 - r166416;
        double r166418 = cbrt(r166415);
        double r166419 = r166418 * r166418;
        double r166420 = r166402 / r166419;
        double r166421 = r166407 / r166418;
        double r166422 = r166420 * r166421;
        double r166423 = log(r166406);
        double r166424 = r166423 / r166415;
        double r166425 = fma(r166406, r166422, r166424);
        double r166426 = r166405 - r166425;
        double r166427 = r166404 ? r166417 : r166426;
        return r166427;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.3
Target16.6
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.184725816409017e-08

    1. Initial program 11.8

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

    3. Applied add-cube-cbrt11.8

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

    4. Applied fma-def11.8

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

    if -5.184725816409017e-08 < z

    1. Initial program 31.2

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

    2. Taylor expanded around 0 7.5

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

    3. Simplified7.5

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

    4. Taylor expanded around 0 7.6

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

    5. Simplified7.6

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

    7. Applied add-cube-cbrt7.8

      \[\leadsto x - \mathsf{fma}\left(1, \frac{z \cdot y}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}, \frac{\log 1}{t}\right)\]

    8. Applied times-frac7.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.184725816409017 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}, \sqrt[3]{1 - y}, y \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}, \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))