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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.30675114307191926 \cdot 10^{-79}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \le -2.30675114307191926 \cdot 10^{-79}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}\\

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

\end{array}

double f(double x, double y, double z, double t) {
        double r311766 = x;
        double r311767 = 1.0;
        double r311768 = y;
        double r311769 = r311767 - r311768;
        double r311770 = z;
        double r311771 = exp(r311770);
        double r311772 = r311768 * r311771;
        double r311773 = r311769 + r311772;
        double r311774 = log(r311773);
        double r311775 = t;
        double r311776 = r311774 / r311775;
        double r311777 = r311766 - r311776;
        return r311777;
}

↓

double f(double x, double y, double z, double t) {
        double r311778 = z;
        double r311779 = -2.3067511430719193e-79;
        bool r311780 = r311778 <= r311779;
        double r311781 = x;
        double r311782 = 1.0;
        double r311783 = t;
        double r311784 = expm1(r311778);
        double r311785 = y;
        double r311786 = 1.0;
        double r311787 = fma(r311784, r311785, r311786);
        double r311788 = log(r311787);
        double r311789 = r311783 / r311788;
        double r311790 = r311782 / r311789;
        double r311791 = r311781 - r311790;
        double r311792 = 0.5;
        double r311793 = 2.0;
        double r311794 = pow(r311778, r311793);
        double r311795 = r311794 * r311785;
        double r311796 = r311778 * r311785;
        double r311797 = log(r311786);
        double r311798 = fma(r311786, r311796, r311797);
        double r311799 = fma(r311792, r311795, r311798);
        double r311800 = r311799 / r311783;
        double r311801 = r311781 - r311800;
        double r311802 = r311780 ? r311791 : r311801;
        return r311802;
}

Original	24.7
Target	16.3
Herbie	8.5

Derivation

Split input into 2 regimes
if z < -2.3067511430719193e-79
1. Initial program 14.9
  \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified11.5
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}}\]
3. Using strategy rm
4. Applied clear-num11.6
  \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}}\]
if -2.3067511430719193e-79 < z
1. Initial program 30.9
  \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified11.4
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}}\]
3. Taylor expanded around 0 6.5
  \[\leadsto x - \frac{\color{blue}{0.5 \cdot \left({z}^{2} \cdot y\right) + \left(1 \cdot \left(z \cdot y\right) + \log 1\right)}}{t}\]
4. Simplified6.5
  \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}}{t}\]
Recombined 2 regimes into one program.
Final simplification8.5
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.30675114307191926 \cdot 10^{-79}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))

Falling back on racket egraph because egg-herbie package not installed (more)

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

Error

Target

Derivation

`if z < -2.3067511430719193e-79`

`if -2.3067511430719193e-79 < z`

Reproduce