Average Error: 25.0 → 8.5
Time: 9.6s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.2580599270527733 \cdot 10^{-16}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{elif}\;z \le 1.0673077358209968 \cdot 10^{-59}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{\frac{1}{{z}^{2}}}, \mathsf{fma}\left(\frac{z}{1}, y, \log \left({\left(\sqrt{1}\right)}^{2}\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.2580599270527733 \cdot 10^{-16}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\

\mathbf{elif}\;z \le 1.0673077358209968 \cdot 10^{-59}:\\
\;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{\frac{1}{{z}^{2}}}, \mathsf{fma}\left(\frac{z}{1}, y, \log \left({\left(\sqrt{1}\right)}^{2}\right)\right)\right)}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r297740 = x;
        double r297741 = 1.0;
        double r297742 = y;
        double r297743 = r297741 - r297742;
        double r297744 = z;
        double r297745 = exp(r297744);
        double r297746 = r297742 * r297745;
        double r297747 = r297743 + r297746;
        double r297748 = log(r297747);
        double r297749 = t;
        double r297750 = r297748 / r297749;
        double r297751 = r297740 - r297750;
        return r297751;
}
↓
double f(double x, double y, double z, double t) {
        double r297752 = z;
        double r297753 = -3.2580599270527733e-16;
        bool r297754 = r297752 <= r297753;
        double r297755 = x;
        double r297756 = 1.0;
        double r297757 = t;
        double r297758 = 1.0;
        double r297759 = y;
        double r297760 = r297758 - r297759;
        double r297761 = exp(r297752);
        double r297762 = r297759 * r297761;
        double r297763 = r297760 + r297762;
        double r297764 = log(r297763);
        double r297765 = r297757 / r297764;
        double r297766 = r297756 / r297765;
        double r297767 = r297755 - r297766;
        double r297768 = 1.0673077358209968e-59;
        bool r297769 = r297752 <= r297768;
        double r297770 = 0.5;
        double r297771 = 2.0;
        double r297772 = pow(r297752, r297771);
        double r297773 = r297758 / r297772;
        double r297774 = r297759 / r297773;
        double r297775 = r297752 / r297758;
        double r297776 = sqrt(r297758);
        double r297777 = pow(r297776, r297771);
        double r297778 = log(r297777);
        double r297779 = fma(r297775, r297759, r297778);
        double r297780 = fma(r297770, r297774, r297779);
        double r297781 = r297780 / r297757;
        double r297782 = r297755 - r297781;
        double r297783 = r297772 * r297759;
        double r297784 = fma(r297752, r297759, r297758);
        double r297785 = fma(r297770, r297783, r297784);
        double r297786 = log(r297785);
        double r297787 = r297786 / r297757;
        double r297788 = r297755 - r297787;
        double r297789 = r297769 ? r297782 : r297788;
        double r297790 = r297754 ? r297767 : r297789;
        return r297790;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.0
Target16.2
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.2580599270527733e-16

    1. Initial program 12.2

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

    3. Applied clear-num12.2

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

    if -3.2580599270527733e-16 < z < 1.0673077358209968e-59

    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-cbrt25.9

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

    4. Applied add-sqr-sqrt25.9

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

    5. Applied prod-diff25.9

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

    6. Applied associate-+l+25.9

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

    7. Simplified25.9

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, -\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \color{blue}{\mathsf{fma}\left(y, \mathsf{expm1}\left(z\right), y\right)}\right)}{t}\]

    8. Taylor expanded around 0 6.2

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

    9. Simplified6.2

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

    if 1.0673077358209968e-59 < z

    1. Initial program 28.1

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

    2. Taylor expanded around 0 12.7

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

    3. Simplified12.7

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2580599270527733 \cdot 10^{-16}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{elif}\;z \le 1.0673077358209968 \cdot 10^{-59}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{\frac{1}{{z}^{2}}}, \mathsf{fma}\left(\frac{z}{1}, y, \log \left({\left(\sqrt{1}\right)}^{2}\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))