Average Error: 25.1 → 8.4
Time: 7.4s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.63783683020029204 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.63783683020029204 \cdot 10^{-19}:\\
\;\;\;\;x - \frac{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \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)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r263961 = x;
        double r263962 = 1.0;
        double r263963 = y;
        double r263964 = r263962 - r263963;
        double r263965 = z;
        double r263966 = exp(r263965);
        double r263967 = r263963 * r263966;
        double r263968 = r263964 + r263967;
        double r263969 = log(r263968);
        double r263970 = t;
        double r263971 = r263969 / r263970;
        double r263972 = r263961 - r263971;
        return r263972;
}
↓
double f(double x, double y, double z, double t) {
        double r263973 = z;
        double r263974 = -1.637836830200292e-19;
        bool r263975 = r263973 <= r263974;
        double r263976 = x;
        double r263977 = 1.0;
        double r263978 = y;
        double r263979 = expm1(r263973);
        double r263980 = r263978 * r263979;
        double r263981 = r263977 + r263980;
        double r263982 = log(r263981);
        double r263983 = t;
        double r263984 = r263982 / r263983;
        double r263985 = r263976 - r263984;
        double r263986 = r263973 * r263978;
        double r263987 = r263986 / r263983;
        double r263988 = 0.5;
        double r263989 = 2.0;
        double r263990 = pow(r263973, r263989);
        double r263991 = r263990 * r263978;
        double r263992 = r263991 / r263983;
        double r263993 = log(r263977);
        double r263994 = r263993 / r263983;
        double r263995 = fma(r263988, r263992, r263994);
        double r263996 = fma(r263987, r263977, r263995);
        double r263997 = r263976 - r263996;
        double r263998 = r263975 ? r263985 : r263997;
        return r263998;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.1
Target16.3
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.637836830200292e-19

    1. Initial program 11.6

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

    3. Applied sub-neg11.6

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

    4. Applied associate-+l+11.5

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

    5. Simplified11.3

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \mathsf{expm1}\left(z\right)}\right)}{t}\]

    if -1.637836830200292e-19 < z

    1. Initial program 31.4

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

    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

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.63783683020029204 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\frac{z \cdot y}{t}, 1, \mathsf{fma}\left(0.5, \frac{{z}^{2} \cdot y}{t}, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))