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

Average Error: 25.6 → 9.3

Time: 19.7s

Precision: 64

• could not determine a ground truth for program body

  1. x = -2.8134698785925108e-55
  2. y = 3520095067841713.5
  3. z = 9.894543499895054e+175
  4. t = 1.5814205469702973e+197

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.731260742038221807671652239454073553396 \cdot 10^{-171}:\\ \;\;\;\;x - \frac{\left(\log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right) + \log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right)\right) + \left(\log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right) + \log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z \cdot y}{t}, \mathsf{fma}\left(\frac{{z}^{2} \cdot y}{t}, 0.5, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r172278 = x;
        double r172279 = 1.0;
        double r172280 = y;
        double r172281 = r172279 - r172280;
        double r172282 = z;
        double r172283 = exp(r172282);
        double r172284 = r172280 * r172283;
        double r172285 = r172281 + r172284;
        double r172286 = log(r172285);
        double r172287 = t;
        double r172288 = r172286 / r172287;
        double r172289 = r172278 - r172288;
        return r172289;
}

↓

double f(double x, double y, double z, double t) {
        double r172290 = z;
        double r172291 = -3.731260742038222e-171;
        bool r172292 = r172290 <= r172291;
        double r172293 = x;
        double r172294 = expm1(r172290);
        double r172295 = y;
        double r172296 = 1.0;
        double r172297 = fma(r172294, r172295, r172296);
        double r172298 = sqrt(r172297);
        double r172299 = sqrt(r172298);
        double r172300 = log(r172299);
        double r172301 = r172300 + r172300;
        double r172302 = r172301 + r172301;
        double r172303 = t;
        double r172304 = r172302 / r172303;
        double r172305 = r172293 - r172304;
        double r172306 = r172290 * r172295;
        double r172307 = r172306 / r172303;
        double r172308 = 2.0;
        double r172309 = pow(r172290, r172308);
        double r172310 = r172309 * r172295;
        double r172311 = r172310 / r172303;
        double r172312 = 0.5;
        double r172313 = log(r172296);
        double r172314 = r172313 / r172303;
        double r172315 = fma(r172311, r172312, r172314);
        double r172316 = fma(r172296, r172307, r172315);
        double r172317 = r172293 - r172316;
        double r172318 = r172292 ? r172305 : r172317;
        return r172318;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.6
Target	16.5
Herbie	9.3

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

Derivation

1. Split input into 2 regimes

2. if z < -3.731260742038222e-171

   1. Initial program 19.3

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

   2. Simplified 12.2

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

   4. Applied add-sqr-sqrt 12.2

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

   5. Applied log-prod 12.2

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

   7. Applied add-sqr-sqrt 12.2

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

   8. Applied sqrt-prod 12.2

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

   9. Applied log-prod 12.2

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

   11. Applied add-sqr-sqrt 12.2

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

   12. Applied sqrt-prod 12.2

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

   13. Applied log-prod 12.2

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

   if -3.731260742038222e-171 < z

   1. Initial program 31.4

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

   2. Simplified 11.6

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

   3. Taylor expanded around 0 6.7

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

   4. Simplified 6.7

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.731260742038221807671652239454073553396 \cdot 10^{-171}:\\ \;\;\;\;x - \frac{\left(\log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right) + \log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right)\right) + \left(\log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right) + \log \left(\sqrt{\sqrt{\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)}}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, \frac{z \cdot y}{t}, \mathsf{fma}\left(\frac{{z}^{2} \cdot y}{t}, 0.5, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))