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

Average Error: 25.0 → 7.9

Time: 32.7s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.611241042596544e-104
  2. y = 4.830431075877442e-71
  3. z = 7.714959714311575e+88
  4. t = -1.8254437613711665e+160

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -0.001725048177624938488400330172112262516748:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(\frac{\log 1}{t} + \left(y \cdot \frac{z}{t}\right) \cdot 1\right) + \left(\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \left(y \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}}}\right)\right) \cdot \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot 0.5\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r17156082 = x;
        double r17156083 = 1.0;
        double r17156084 = y;
        double r17156085 = r17156083 - r17156084;
        double r17156086 = z;
        double r17156087 = exp(r17156086);
        double r17156088 = r17156084 * r17156087;
        double r17156089 = r17156085 + r17156088;
        double r17156090 = log(r17156089);
        double r17156091 = t;
        double r17156092 = r17156090 / r17156091;
        double r17156093 = r17156082 - r17156092;
        return r17156093;
}

↓

double f(double x, double y, double z, double t) {
        double r17156094 = z;
        double r17156095 = -0.0017250481776249385;
        bool r17156096 = r17156094 <= r17156095;
        double r17156097 = x;
        double r17156098 = exp(r17156094);
        double r17156099 = y;
        double r17156100 = r17156098 * r17156099;
        double r17156101 = 1.0;
        double r17156102 = r17156101 - r17156099;
        double r17156103 = r17156100 + r17156102;
        double r17156104 = log(r17156103);
        double r17156105 = 1.0;
        double r17156106 = t;
        double r17156107 = r17156105 / r17156106;
        double r17156108 = r17156104 * r17156107;
        double r17156109 = r17156097 - r17156108;
        double r17156110 = log(r17156101);
        double r17156111 = r17156110 / r17156106;
        double r17156112 = r17156094 / r17156106;
        double r17156113 = r17156099 * r17156112;
        double r17156114 = r17156113 * r17156101;
        double r17156115 = r17156111 + r17156114;
        double r17156116 = cbrt(r17156094);
        double r17156117 = r17156116 * r17156116;
        double r17156118 = cbrt(r17156106);
        double r17156119 = cbrt(r17156118);
        double r17156120 = r17156119 * r17156119;
        double r17156121 = r17156117 / r17156120;
        double r17156122 = r17156116 / r17156119;
        double r17156123 = r17156099 * r17156122;
        double r17156124 = r17156121 * r17156123;
        double r17156125 = r17156118 * r17156118;
        double r17156126 = r17156094 / r17156125;
        double r17156127 = r17156124 * r17156126;
        double r17156128 = 0.5;
        double r17156129 = r17156127 * r17156128;
        double r17156130 = r17156115 + r17156129;
        double r17156131 = r17156097 - r17156130;
        double r17156132 = r17156096 ? r17156109 : r17156131;
        return r17156132;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.0
Target	16.4
Herbie	7.9

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

Derivation

1. Split input into 2 regimes

2. if z < -0.0017250481776249385

   1. Initial program 11.1

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

   3. Applied div-inv 11.1

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

   if -0.0017250481776249385 < z

   1. Initial program 31.1

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

   2. Taylor expanded around 0 7.4

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

   3. Simplified 6.3

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

   5. Applied add-cube-cbrt 6.3

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

   6. Applied times-frac 6.3

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

   7. Applied associate-*l* 6.4

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

   9. Applied add-cube-cbrt 6.4

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

   10. Applied add-cube-cbrt 6.4

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

   11. Applied times-frac 6.4

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

   12. Applied associate-*l* 6.4

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.001725048177624938488400330172112262516748:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(\frac{\log 1}{t} + \left(y \cdot \frac{z}{t}\right) \cdot 1\right) + \left(\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}} \cdot \left(y \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{t}}}\right)\right) \cdot \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))