Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Average Error: 17.8 → 0.1

Time: 14.9s

Precision: 64

Math

\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -61158959.90174198150634765625 \lor \neg \left(y \le 33791339.152441658079624176025390625\right):\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r264402 = 1.0;
        double r264403 = x;
        double r264404 = y;
        double r264405 = r264403 - r264404;
        double r264406 = r264402 - r264404;
        double r264407 = r264405 / r264406;
        double r264408 = r264402 - r264407;
        double r264409 = log(r264408);
        double r264410 = r264402 - r264409;
        return r264410;
}

↓

double f(double x, double y) {
        double r264411 = y;
        double r264412 = -61158959.90174198;
        bool r264413 = r264411 <= r264412;
        double r264414 = 33791339.15244166;
        bool r264415 = r264411 <= r264414;
        double r264416 = !r264415;
        bool r264417 = r264413 || r264416;
        double r264418 = 1.0;
        double r264419 = r264418 / r264411;
        double r264420 = 1.0;
        double r264421 = r264419 + r264420;
        double r264422 = x;
        double r264423 = r264422 / r264411;
        double r264424 = r264421 * r264423;
        double r264425 = r264424 - r264419;
        double r264426 = log(r264425);
        double r264427 = r264418 - r264426;
        double r264428 = r264418 - r264411;
        double r264429 = cbrt(r264428);
        double r264430 = r264429 * r264429;
        double r264431 = r264420 / r264430;
        double r264432 = r264422 - r264411;
        double r264433 = r264432 / r264429;
        double r264434 = r264431 * r264433;
        double r264435 = r264418 - r264434;
        double r264436 = log(r264435);
        double r264437 = r264418 - r264436;
        double r264438 = r264417 ? r264427 : r264437;
        return r264438;
}

Error

Bits error versus x

Bits error versus y

Target

Original	17.8
Target	0.1
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -61158959.90174198 or 33791339.15244166 < y

   1. Initial program 46.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

   if -61158959.90174198 < y < 33791339.15244166

   1. Initial program 0.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.1

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

   4. Applied *-un-lft-identity 0.1

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

   5. Applied times-frac 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -61158959.90174198150634765625 \lor \neg \left(y \le 33791339.152441658079624176025390625\right):\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + 1\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))