Average Error: 25.2 → 9.0
Time: 8.1s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.086785831759009406096340959265535489196 \cdot 10^{-25}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{elif}\;z \le 2.155036395743891297675849135471205071901 \cdot 10^{-100}:\\ \;\;\;\;x - \left(1 \cdot \frac{1}{\frac{t}{z \cdot y}} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.086785831759009406096340959265535489196 \cdot 10^{-25}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\

\mathbf{elif}\;z \le 2.155036395743891297675849135471205071901 \cdot 10^{-100}:\\
\;\;\;\;x - \left(1 \cdot \frac{1}{\frac{t}{z \cdot y}} + \frac{\log 1}{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r276144 = x;
        double r276145 = 1.0;
        double r276146 = y;
        double r276147 = r276145 - r276146;
        double r276148 = z;
        double r276149 = exp(r276148);
        double r276150 = r276146 * r276149;
        double r276151 = r276147 + r276150;
        double r276152 = log(r276151);
        double r276153 = t;
        double r276154 = r276152 / r276153;
        double r276155 = r276144 - r276154;
        return r276155;
}
↓
double f(double x, double y, double z, double t) {
        double r276156 = z;
        double r276157 = -2.0867858317590094e-25;
        bool r276158 = r276156 <= r276157;
        double r276159 = x;
        double r276160 = 1.0;
        double r276161 = y;
        double r276162 = r276160 - r276161;
        double r276163 = exp(r276156);
        double r276164 = r276161 * r276163;
        double r276165 = cbrt(r276164);
        double r276166 = r276165 * r276165;
        double r276167 = r276166 * r276165;
        double r276168 = r276162 + r276167;
        double r276169 = log(r276168);
        double r276170 = t;
        double r276171 = r276169 / r276170;
        double r276172 = r276159 - r276171;
        double r276173 = 2.1550363957438913e-100;
        bool r276174 = r276156 <= r276173;
        double r276175 = 1.0;
        double r276176 = r276156 * r276161;
        double r276177 = r276170 / r276176;
        double r276178 = r276175 / r276177;
        double r276179 = r276160 * r276178;
        double r276180 = log(r276160);
        double r276181 = r276180 / r276170;
        double r276182 = r276179 + r276181;
        double r276183 = r276159 - r276182;
        double r276184 = 0.5;
        double r276185 = 2.0;
        double r276186 = pow(r276156, r276185);
        double r276187 = r276184 * r276186;
        double r276188 = r276187 + r276156;
        double r276189 = r276161 * r276188;
        double r276190 = r276160 + r276189;
        double r276191 = log(r276190);
        double r276192 = r276191 / r276170;
        double r276193 = r276159 - r276192;
        double r276194 = r276174 ? r276183 : r276193;
        double r276195 = r276158 ? r276172 : r276194;
        return r276195;
}

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

Original25.2
Target16.8
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.0867858317590094e-25

    1. Initial program 13.1

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

    3. Applied add-cube-cbrt12.9

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

    if -2.0867858317590094e-25 < z < 2.1550363957438913e-100

    1. Initial program 31.3

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

    3. Applied add-cube-cbrt25.8

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

    4. Taylor expanded around 0 5.7

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

    5. Simplified5.7

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

    6. Taylor expanded around 0 5.7

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

    8. Applied clear-num5.7

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

    if 2.1550363957438913e-100 < z

    1. Initial program 29.7

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

    2. Taylor expanded around 0 14.6

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

    3. Simplified14.6

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.086785831759009406096340959265535489196 \cdot 10^{-25}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{elif}\;z \le 2.155036395743891297675849135471205071901 \cdot 10^{-100}:\\ \;\;\;\;x - \left(1 \cdot \frac{1}{\frac{t}{z \cdot y}} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))