System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Average Error: 25.1 → 9.4

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -71733.55053223111:\\ \;\;\;\;x - \frac{2 \cdot \log \left(\sqrt[3]{\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}}\right) + \log \left(\sqrt[3]{\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}}\right)}{t}\\ \mathbf{elif}\;z \le 5.1398477853792008 \cdot 10^{-188}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r308782 = x;
        double r308783 = 1.0;
        double r308784 = y;
        double r308785 = r308783 - r308784;
        double r308786 = z;
        double r308787 = exp(r308786);
        double r308788 = r308784 * r308787;
        double r308789 = r308785 + r308788;
        double r308790 = log(r308789);
        double r308791 = t;
        double r308792 = r308790 / r308791;
        double r308793 = r308782 - r308792;
        return r308793;
}

↓

double f(double x, double y, double z, double t) {
        double r308794 = z;
        double r308795 = -71733.55053223111;
        bool r308796 = r308794 <= r308795;
        double r308797 = x;
        double r308798 = 2.0;
        double r308799 = 1.0;
        double r308800 = y;
        double r308801 = r308799 - r308800;
        double r308802 = exp(r308794);
        double r308803 = r308800 * r308802;
        double r308804 = cbrt(r308803);
        double r308805 = r308804 * r308804;
        double r308806 = r308805 * r308804;
        double r308807 = r308801 + r308806;
        double r308808 = cbrt(r308807);
        double r308809 = log(r308808);
        double r308810 = r308798 * r308809;
        double r308811 = r308810 + r308809;
        double r308812 = t;
        double r308813 = r308811 / r308812;
        double r308814 = r308797 - r308813;
        double r308815 = 5.139847785379201e-188;
        bool r308816 = r308794 <= r308815;
        double r308817 = log(r308799);
        double r308818 = 0.5;
        double r308819 = pow(r308794, r308798);
        double r308820 = r308818 * r308819;
        double r308821 = r308799 * r308794;
        double r308822 = r308820 + r308821;
        double r308823 = r308800 * r308822;
        double r308824 = r308817 + r308823;
        double r308825 = r308824 / r308812;
        double r308826 = r308797 - r308825;
        double r308827 = 0.5;
        double r308828 = r308827 * r308819;
        double r308829 = r308828 + r308794;
        double r308830 = r308800 * r308829;
        double r308831 = r308799 + r308830;
        double r308832 = log(r308831);
        double r308833 = r308832 / r308812;
        double r308834 = r308797 - r308833;
        double r308835 = r308816 ? r308826 : r308834;
        double r308836 = r308796 ? r308814 : r308835;
        return r308836;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.1
Target	16.8
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if z < -71733.55053223111

   1. Initial program 12.1

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

   3. Applied add-cube-cbrt 12.1

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

   5. Applied add-cube-cbrt 12.2

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

   6. Applied log-prod 12.2

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

   7. Simplified 12.2

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

   if -71733.55053223111 < z < 5.139847785379201e-188

   1. Initial program 30.5

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

   3. Applied add-cube-cbrt 25.8

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

   4. Taylor expanded around 0 6.5

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

   5. Simplified 6.5

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

   if 5.139847785379201e-188 < z

   1. Initial program 30.5

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

   2. Taylor expanded around 0 12.4

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

   3. Simplified 12.4

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -71733.55053223111:\\ \;\;\;\;x - \frac{2 \cdot \log \left(\sqrt[3]{\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}}\right) + \log \left(\sqrt[3]{\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}}\right)}{t}\\ \mathbf{elif}\;z \le 5.1398477853792008 \cdot 10^{-188}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))