Average Error: 25.5 → 9.2
Time: 7.0s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\
\;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\

\mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\
\;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r293173 = x;
        double r293174 = 1.0;
        double r293175 = y;
        double r293176 = r293174 - r293175;
        double r293177 = z;
        double r293178 = exp(r293177);
        double r293179 = r293175 * r293178;
        double r293180 = r293176 + r293179;
        double r293181 = log(r293180);
        double r293182 = t;
        double r293183 = r293181 / r293182;
        double r293184 = r293173 - r293183;
        return r293184;
}
↓
double f(double x, double y, double z, double t) {
        double r293185 = z;
        double r293186 = -4.2011862993899546e-06;
        bool r293187 = r293185 <= r293186;
        double r293188 = x;
        double r293189 = 1.0;
        double r293190 = exp(r293185);
        double r293191 = 1.0;
        double r293192 = r293190 - r293191;
        double r293193 = y;
        double r293194 = r293192 * r293193;
        double r293195 = r293189 + r293194;
        double r293196 = log(r293195);
        double r293197 = t;
        double r293198 = r293196 / r293197;
        double r293199 = r293188 - r293198;
        double r293200 = 3.4446276387824773e-225;
        bool r293201 = r293185 <= r293200;
        double r293202 = log(r293189);
        double r293203 = 0.5;
        double r293204 = 2.0;
        double r293205 = pow(r293185, r293204);
        double r293206 = r293203 * r293205;
        double r293207 = r293189 * r293185;
        double r293208 = r293206 + r293207;
        double r293209 = r293193 * r293208;
        double r293210 = r293202 + r293209;
        double r293211 = r293210 / r293197;
        double r293212 = r293188 - r293211;
        double r293213 = 0.16666666666666666;
        double r293214 = 3.0;
        double r293215 = pow(r293185, r293214);
        double r293216 = r293213 * r293215;
        double r293217 = 0.5;
        double r293218 = r293217 * r293205;
        double r293219 = r293216 + r293218;
        double r293220 = r293185 + r293219;
        double r293221 = r293193 * r293220;
        double r293222 = r293189 + r293221;
        double r293223 = log(r293222);
        double r293224 = r293223 / r293197;
        double r293225 = r293188 - r293224;
        double r293226 = r293201 ? r293212 : r293225;
        double r293227 = r293187 ? r293199 : r293226;
        return r293227;
}

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

Original25.5
Target16.5
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.2011862993899546e-06

    1. Initial program 11.9

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

    3. Applied sub-neg11.9

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

    4. Applied associate-+l+11.9

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

    5. Simplified11.9

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

    if -4.2011862993899546e-06 < z < 3.4446276387824773e-225

    1. Initial program 31.7

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

    2. Taylor expanded around 0 6.1

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

    3. Simplified6.1

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

    if 3.4446276387824773e-225 < z

    1. Initial program 31.1

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

    3. Applied sub-neg31.1

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

    4. Applied associate-+l+18.1

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

    5. Simplified18.1

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

    6. Taylor expanded around 0 11.5

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left(\frac{1}{6} \cdot \left({z}^{3} \cdot y\right) + \left(z \cdot y + \frac{1}{2} \cdot \left({z}^{2} \cdot y\right)\right)\right)}\right)}{t}\]

    7. Simplified11.5

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)}\right)}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))