Average Error: 25.2 → 8.5
Time: 9.9s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\left(z \cdot y\right) \cdot \frac{1}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.0:\\
\;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r277616 = x;
        double r277617 = 1.0;
        double r277618 = y;
        double r277619 = r277617 - r277618;
        double r277620 = z;
        double r277621 = exp(r277620);
        double r277622 = r277618 * r277621;
        double r277623 = r277619 + r277622;
        double r277624 = log(r277623);
        double r277625 = t;
        double r277626 = r277624 / r277625;
        double r277627 = r277616 - r277626;
        return r277627;
}
↓
double f(double x, double y, double z, double t) {
        double r277628 = z;
        double r277629 = exp(r277628);
        double r277630 = 0.0;
        bool r277631 = r277629 <= r277630;
        double r277632 = x;
        double r277633 = 1.0;
        double r277634 = y;
        double r277635 = r277633 - r277634;
        double r277636 = r277634 * r277629;
        double r277637 = r277635 + r277636;
        double r277638 = log(r277637);
        double r277639 = 1.0;
        double r277640 = t;
        double r277641 = r277639 / r277640;
        double r277642 = r277638 * r277641;
        double r277643 = r277632 - r277642;
        double r277644 = r277628 * r277634;
        double r277645 = r277644 * r277641;
        double r277646 = 0.5;
        double r277647 = 2.0;
        double r277648 = pow(r277628, r277647);
        double r277649 = r277648 * r277634;
        double r277650 = r277649 / r277640;
        double r277651 = log(r277633);
        double r277652 = r277651 / r277640;
        double r277653 = fma(r277646, r277650, r277652);
        double r277654 = fma(r277645, r277633, r277653);
        double r277655 = r277632 - r277654;
        double r277656 = r277631 ? r277643 : r277655;
        return r277656;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.2
Target16.6
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.0

    1. Initial program 11.5

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

    3. Applied div-inv11.5

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

    if 0.0 < (exp 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 7.1

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

    3. Simplified7.1

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

    5. Applied div-inv7.2

      \[\leadsto x - \mathsf{fma}\left(\color{blue}{\left(z \cdot y\right) \cdot \frac{1}{t}}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\left(z \cdot y\right) \cdot \frac{1}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))