\[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.6516919137036032 \cdot 10^{-24}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1.0\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{\left(0.5 \cdot z\right) \cdot z}{t}, y, \mathsf{fma}\left(\frac{z}{t}, y \cdot 1.0, \frac{\log 1.0}{t}\right)\right)\\ \end{array}\]

x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}

↓

\begin{array}{l}
\mathbf{if}\;z \le -2.6516919137036032 \cdot 10^{-24}:\\
\;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1.0\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \mathsf{fma}\left(\frac{\left(0.5 \cdot z\right) \cdot z}{t}, y, \mathsf{fma}\left(\frac{z}{t}, y \cdot 1.0, \frac{\log 1.0}{t}\right)\right)\\

\end{array}

double f(double x, double y, double z, double t) {
        double r13536941 = x;
        double r13536942 = 1.0;
        double r13536943 = y;
        double r13536944 = r13536942 - r13536943;
        double r13536945 = z;
        double r13536946 = exp(r13536945);
        double r13536947 = r13536943 * r13536946;
        double r13536948 = r13536944 + r13536947;
        double r13536949 = log(r13536948);
        double r13536950 = t;
        double r13536951 = r13536949 / r13536950;
        double r13536952 = r13536941 - r13536951;
        return r13536952;
}

↓

double f(double x, double y, double z, double t) {
        double r13536953 = z;
        double r13536954 = -2.6516919137036032e-24;
        bool r13536955 = r13536953 <= r13536954;
        double r13536956 = x;
        double r13536957 = expm1(r13536953);
        double r13536958 = y;
        double r13536959 = 1.0;
        double r13536960 = fma(r13536957, r13536958, r13536959);
        double r13536961 = log(r13536960);
        double r13536962 = t;
        double r13536963 = r13536961 / r13536962;
        double r13536964 = r13536956 - r13536963;
        double r13536965 = 0.5;
        double r13536966 = r13536965 * r13536953;
        double r13536967 = r13536966 * r13536953;
        double r13536968 = r13536967 / r13536962;
        double r13536969 = r13536953 / r13536962;
        double r13536970 = r13536958 * r13536959;
        double r13536971 = log(r13536959);
        double r13536972 = r13536971 / r13536962;
        double r13536973 = fma(r13536969, r13536970, r13536972);
        double r13536974 = fma(r13536968, r13536958, r13536973);
        double r13536975 = r13536956 - r13536974;
        double r13536976 = r13536955 ? r13536964 : r13536975;
        return r13536976;
}

Original	25.0
Target	16.3
Herbie	8.0

Derivation

Split input into 2 regimes
if z < -2.6516919137036032e-24
1. Initial program 12.1
  \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified11.6
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1.0\right)\right)}{t}}\]
if -2.6516919137036032e-24 < z
1. Initial program 30.9
  \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
2. Simplified11.6
  \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1.0\right)\right)}{t}}\]
3. Taylor expanded around 0 7.1
  \[\leadsto x - \color{blue}{\left(1.0 \cdot \frac{z \cdot y}{t} + \left(0.5 \cdot \frac{{z}^{2} \cdot y}{t} + \frac{\log 1.0}{t}\right)\right)}\]
4. Simplified6.4
  \[\leadsto x - \color{blue}{\mathsf{fma}\left(\frac{\left(0.5 \cdot z\right) \cdot z}{t}, y, \mathsf{fma}\left(\frac{z}{t}, y \cdot 1.0, \frac{\log 1.0}{t}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification8.0
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.6516919137036032 \cdot 10^{-24}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1.0\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{\left(0.5 \cdot z\right) \cdot z}{t}, y, \mathsf{fma}\left(\frac{z}{t}, y \cdot 1.0, \frac{\log 1.0}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))

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

Error

Target

Derivation

`if z < -2.6516919137036032e-24`

`if -2.6516919137036032e-24 < z`

Reproduce