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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\sqrt[3]{{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\right)}^{3}}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z \cdot y}{t}, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

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 - \frac{\sqrt[3]{{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\right)}^{3}}}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \mathsf{fma}\left(1, \frac{z \cdot y}{t}, \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 r279667 = x;
        double r279668 = 1.0;
        double r279669 = y;
        double r279670 = r279668 - r279669;
        double r279671 = z;
        double r279672 = exp(r279671);
        double r279673 = r279669 * r279672;
        double r279674 = r279670 + r279673;
        double r279675 = log(r279674);
        double r279676 = t;
        double r279677 = r279675 / r279676;
        double r279678 = r279667 - r279677;
        return r279678;
}

↓

double f(double x, double y, double z, double t) {
        double r279679 = z;
        double r279680 = exp(r279679);
        double r279681 = 0.0;
        bool r279682 = r279680 <= r279681;
        double r279683 = x;
        double r279684 = expm1(r279679);
        double r279685 = y;
        double r279686 = 1.0;
        double r279687 = fma(r279684, r279685, r279686);
        double r279688 = log(r279687);
        double r279689 = 3.0;
        double r279690 = pow(r279688, r279689);
        double r279691 = cbrt(r279690);
        double r279692 = t;
        double r279693 = r279691 / r279692;
        double r279694 = r279683 - r279693;
        double r279695 = r279679 * r279685;
        double r279696 = r279695 / r279692;
        double r279697 = 0.5;
        double r279698 = 2.0;
        double r279699 = pow(r279679, r279698);
        double r279700 = r279699 * r279685;
        double r279701 = r279700 / r279692;
        double r279702 = log(r279686);
        double r279703 = r279702 / r279692;
        double r279704 = fma(r279697, r279701, r279703);
        double r279705 = fma(r279686, r279696, r279704);
        double r279706 = r279683 - r279705;
        double r279707 = r279682 ? r279694 : r279706;
        return r279707;
}

Original	25.3
Target	16.5
Herbie	8.9

Derivation

Split input into 2 regimes
if (exp z) < 0.0
1. Initial program 12.1
  \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified12.1
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}}\]
3. Using strategy rm
4. Applied add-cbrt-cube12.1
  \[\leadsto x - \frac{\color{blue}{\sqrt[3]{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right) \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\right) \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}}{t}\]
5. Simplified12.1
  \[\leadsto x - \frac{\sqrt[3]{\color{blue}{{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\right)}^{3}}}}{t}\]
if 0.0 < (exp z)
1. Initial program 30.8
  \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified11.7
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}}\]
3. Taylor expanded around 0 7.5
  \[\leadsto x - \color{blue}{\left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)}\]
4. Simplified7.5
  \[\leadsto x - \color{blue}{\mathsf{fma}\left(1, \frac{z \cdot y}{t}, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification8.9
\[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\sqrt[3]{{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)\right)}^{3}}}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z \cdot y}{t}, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))

could not determine a ground truth for program body (more)

Error

Target

Derivation

`if (exp z) < 0.0`

`if 0.0 < (exp z)`

Reproduce