Average Error: 25.5 → 9.3
Time: 8.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.61247754966531634 \cdot 10^{-6}:\\ \;\;\;\;x - \log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.61247754966531634 \cdot 10^{-6}:\\
\;\;\;\;x - \log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right) \cdot \frac{1}{t}\\

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

\mathbf{else}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r338579 = x;
        double r338580 = 1.0;
        double r338581 = y;
        double r338582 = r338580 - r338581;
        double r338583 = z;
        double r338584 = exp(r338583);
        double r338585 = r338581 * r338584;
        double r338586 = r338582 + r338585;
        double r338587 = log(r338586);
        double r338588 = t;
        double r338589 = r338587 / r338588;
        double r338590 = r338579 - r338589;
        return r338590;
}
↓
double f(double x, double y, double z, double t) {
        double r338591 = z;
        double r338592 = -3.6124775496653163e-06;
        bool r338593 = r338591 <= r338592;
        double r338594 = x;
        double r338595 = 1.0;
        double r338596 = y;
        double r338597 = expm1(r338591);
        double r338598 = r338596 * r338597;
        double r338599 = r338595 + r338598;
        double r338600 = log(r338599);
        double r338601 = 1.0;
        double r338602 = t;
        double r338603 = r338601 / r338602;
        double r338604 = r338600 * r338603;
        double r338605 = r338594 - r338604;
        double r338606 = 3.4446276387824773e-225;
        bool r338607 = r338591 <= r338606;
        double r338608 = 0.5;
        double r338609 = 2.0;
        double r338610 = pow(r338591, r338609);
        double r338611 = r338610 * r338596;
        double r338612 = r338591 * r338596;
        double r338613 = log(r338595);
        double r338614 = fma(r338595, r338612, r338613);
        double r338615 = fma(r338608, r338611, r338614);
        double r338616 = r338615 * r338603;
        double r338617 = r338594 - r338616;
        double r338618 = r338602 / r338600;
        double r338619 = r338601 / r338618;
        double r338620 = r338594 - r338619;
        double r338621 = r338607 ? r338617 : r338620;
        double r338622 = r338593 ? r338605 : r338621;
        return r338622;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.5
Target16.5
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.6124775496653163e-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. Simplified11.8

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

    7. Applied div-inv11.9

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

    if -3.6124775496653163e-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. Using strategy rm

    3. Applied sub-neg31.7

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

    4. Applied associate-+l+16.0

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

    5. Simplified11.9

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

    7. Applied div-inv11.9

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

    8. Taylor expanded around 0 6.1

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

    9. Simplified6.1

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

    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. Simplified11.9

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

    7. Applied clear-num11.9

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.61247754966531634 \cdot 10^{-6}:\\ \;\;\;\;x - \log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(1 + y \cdot \mathsf{expm1}\left(z\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))