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

Average Error: 18.1 → 0.8

Time: 14.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.960614984267581169834128874268630582378 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r312403 = 1.0;
        double r312404 = x;
        double r312405 = y;
        double r312406 = r312404 - r312405;
        double r312407 = r312403 - r312405;
        double r312408 = r312406 / r312407;
        double r312409 = r312403 - r312408;
        double r312410 = log(r312409);
        double r312411 = r312403 - r312410;
        return r312411;
}

↓

double f(double x, double y) {
        double r312412 = x;
        double r312413 = y;
        double r312414 = r312412 - r312413;
        double r312415 = 1.0;
        double r312416 = r312415 - r312413;
        double r312417 = r312414 / r312416;
        double r312418 = 3.960614984267581e-08;
        bool r312419 = r312417 <= r312418;
        double r312420 = 1.0;
        double r312421 = cbrt(r312416);
        double r312422 = r312421 * r312421;
        double r312423 = r312420 / r312422;
        double r312424 = r312414 / r312421;
        double r312425 = r312423 * r312424;
        double r312426 = r312415 - r312425;
        double r312427 = log(r312426);
        double r312428 = r312415 - r312427;
        double r312429 = r312412 / r312413;
        double r312430 = 2.0;
        double r312431 = pow(r312413, r312430);
        double r312432 = r312412 / r312431;
        double r312433 = r312415 * r312432;
        double r312434 = r312429 + r312433;
        double r312435 = r312415 / r312413;
        double r312436 = r312434 - r312435;
        double r312437 = log(r312436);
        double r312438 = r312415 - r312437;
        double r312439 = r312419 ? r312428 : r312438;
        return r312439;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.1
Target	0.1
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 3.960614984267581e-08

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied times-frac 0.0

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

   if 3.960614984267581e-08 < (/ (- x y) (- 1.0 y))

   1. Initial program 58.9

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

   2. Taylor expanded around inf 2.6

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

   3. Simplified 2.6

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.960614984267581169834128874268630582378 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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))))))