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

Average Error: 24.8 → 7.7

Time: 9.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.999999999977131515:\\ \;\;\;\;x - \frac{2 \cdot \log \left(\sqrt[3]{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt[3]{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}} \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right) \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r360973 = x;
        double r360974 = 1.0;
        double r360975 = y;
        double r360976 = r360974 - r360975;
        double r360977 = z;
        double r360978 = exp(r360977);
        double r360979 = r360975 * r360978;
        double r360980 = r360976 + r360979;
        double r360981 = log(r360980);
        double r360982 = t;
        double r360983 = r360981 / r360982;
        double r360984 = r360973 - r360983;
        return r360984;
}

↓

double f(double x, double y, double z, double t) {
        double r360985 = z;
        double r360986 = exp(r360985);
        double r360987 = 0.9999999999771315;
        bool r360988 = r360986 <= r360987;
        double r360989 = x;
        double r360990 = 2.0;
        double r360991 = 1.0;
        double r360992 = y;
        double r360993 = r360991 - r360992;
        double r360994 = r360992 * r360986;
        double r360995 = r360993 + r360994;
        double r360996 = cbrt(r360995);
        double r360997 = log(r360996);
        double r360998 = r360990 * r360997;
        double r360999 = r360998 + r360997;
        double r361000 = t;
        double r361001 = r360999 / r361000;
        double r361002 = r360989 - r361001;
        double r361003 = cbrt(r361000);
        double r361004 = r361003 * r361003;
        double r361005 = r360985 / r361004;
        double r361006 = cbrt(r361003);
        double r361007 = r361006 * r361006;
        double r361008 = r361005 / r361007;
        double r361009 = r360992 / r361006;
        double r361010 = r361008 * r361009;
        double r361011 = cbrt(r361010);
        double r361012 = r361011 * r361011;
        double r361013 = r361012 * r361011;
        double r361014 = r360991 * r361013;
        double r361015 = log(r360991);
        double r361016 = r361015 / r361000;
        double r361017 = 0.5;
        double r361018 = pow(r360985, r360990);
        double r361019 = r361018 * r360992;
        double r361020 = r361019 / r361000;
        double r361021 = r361017 * r361020;
        double r361022 = r361016 + r361021;
        double r361023 = r361014 + r361022;
        double r361024 = r360989 - r361023;
        double r361025 = r360988 ? r361002 : r361024;
        return r361025;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	24.8
Target	15.9
Herbie	7.7

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

Derivation

1. Split input into 2 regimes

2. if (exp z) < 0.9999999999771315

   1. Initial program 10.5

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

   3. Applied add-cube-cbrt 10.5

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

   4. Applied log-prod 10.5

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

   5. Simplified 10.5

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

   if 0.9999999999771315 < (exp z)

   1. Initial program 31.2

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

   2. Taylor expanded around 0 7.1

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

   4. Applied add-cube-cbrt 7.3

      \[\leadsto x - \left(1 \cdot \frac{z \cdot y}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]

   5. Applied times-frac 6.8

      \[\leadsto x - \left(1 \cdot \color{blue}{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right)} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 6.9

      \[\leadsto x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t}}}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]

   8. Applied *-un-lft-identity 6.9

      \[\leadsto x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t}}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]

   9. Applied times-frac 6.9

      \[\leadsto x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}\right)}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]

   10. Applied associate-*r* 6.5

       \[\leadsto x - \left(1 \cdot \color{blue}{\left(\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}\right) \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}\right)} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]

   11. Simplified 6.5

       \[\leadsto x - \left(1 \cdot \left(\color{blue}{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 6.5

       \[\leadsto x - \left(1 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}} \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right) \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right)} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.999999999977131515:\\ \;\;\;\;x - \frac{2 \cdot \log \left(\sqrt[3]{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt[3]{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}} \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right) \cdot \sqrt[3]{\frac{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{t}}}}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))