Average Error: 25.0 → 7.8
Time: 28.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.087859946381865645:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -7.05653233804947833 \cdot 10^{-90}:\\ \;\;\;\;x - \frac{\log \left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + \left(z \cdot y + 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{1}{\frac{\frac{t}{z}}{y}} + \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.087859946381865645:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\

\mathbf{elif}\;z \le -7.05653233804947833 \cdot 10^{-90}:\\
\;\;\;\;x - \frac{\log \left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + \left(z \cdot y + 1\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \left(1 \cdot \frac{1}{\frac{\frac{t}{z}}{y}} + \frac{\log 1}{t}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r239005 = x;
        double r239006 = 1.0;
        double r239007 = y;
        double r239008 = r239006 - r239007;
        double r239009 = z;
        double r239010 = exp(r239009);
        double r239011 = r239007 * r239010;
        double r239012 = r239008 + r239011;
        double r239013 = log(r239012);
        double r239014 = t;
        double r239015 = r239013 / r239014;
        double r239016 = r239005 - r239015;
        return r239016;
}
↓
double f(double x, double y, double z, double t) {
        double r239017 = z;
        double r239018 = -0.08785994638186564;
        bool r239019 = r239017 <= r239018;
        double r239020 = x;
        double r239021 = 1.0;
        double r239022 = y;
        double r239023 = r239021 - r239022;
        double r239024 = cbrt(r239022);
        double r239025 = r239024 * r239024;
        double r239026 = exp(r239017);
        double r239027 = r239024 * r239026;
        double r239028 = r239025 * r239027;
        double r239029 = r239023 + r239028;
        double r239030 = log(r239029);
        double r239031 = t;
        double r239032 = r239030 / r239031;
        double r239033 = r239020 - r239032;
        double r239034 = -7.056532338049478e-90;
        bool r239035 = r239017 <= r239034;
        double r239036 = 0.5;
        double r239037 = 2.0;
        double r239038 = pow(r239017, r239037);
        double r239039 = r239038 * r239022;
        double r239040 = r239036 * r239039;
        double r239041 = r239017 * r239022;
        double r239042 = r239041 + r239021;
        double r239043 = r239040 + r239042;
        double r239044 = log(r239043);
        double r239045 = r239044 / r239031;
        double r239046 = r239020 - r239045;
        double r239047 = 1.0;
        double r239048 = r239031 / r239017;
        double r239049 = r239048 / r239022;
        double r239050 = r239047 / r239049;
        double r239051 = r239021 * r239050;
        double r239052 = log(r239021);
        double r239053 = r239052 / r239031;
        double r239054 = r239051 + r239053;
        double r239055 = r239020 - r239054;
        double r239056 = r239035 ? r239046 : r239055;
        double r239057 = r239019 ? r239033 : r239056;
        return r239057;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.0
Target15.9
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -0.08785994638186564

    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 add-cube-cbrt11.6

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot e^{z}\right)}{t}\]

    4. Applied associate-*l*11.6

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)}\right)}{t}\]

    if -0.08785994638186564 < z < -7.056532338049478e-90

    1. Initial program 28.4

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

    2. Taylor expanded around 0 11.8

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

    if -7.056532338049478e-90 < z

    1. Initial program 30.8

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

    2. Taylor expanded around 0 6.3

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

    3. Simplified6.3

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

    4. Taylor expanded around 0 6.4

      \[\leadsto \color{blue}{x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)}\]
    5. Using strategy rm

    6. Applied clear-num6.4

      \[\leadsto x - \left(1 \cdot \color{blue}{\frac{1}{\frac{t}{z \cdot y}}} + \frac{\log 1}{t}\right)\]
    7. Using strategy rm

    8. Applied associate-/r*5.4

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.087859946381865645:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -7.05653233804947833 \cdot 10^{-90}:\\ \;\;\;\;x - \frac{\log \left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + \left(z \cdot y + 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{1}{\frac{\frac{t}{z}}{y}} + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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)))