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

Average Error: 17.8 → 0.1

Time: 16.0s

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 r393570 = 1.0;
        double r393571 = x;
        double r393572 = y;
        double r393573 = r393571 - r393572;
        double r393574 = r393570 - r393572;
        double r393575 = r393573 / r393574;
        double r393576 = r393570 - r393575;
        double r393577 = log(r393576);
        double r393578 = r393570 - r393577;
        return r393578;
}

↓

double f(double x, double y) {
        double r393579 = y;
        double r393580 = -61158959.90174198;
        bool r393581 = r393579 <= r393580;
        double r393582 = 33791339.15244166;
        bool r393583 = r393579 <= r393582;
        double r393584 = !r393583;
        bool r393585 = r393581 || r393584;
        double r393586 = 1.0;
        double r393587 = r393586 / r393579;
        double r393588 = 1.0;
        double r393589 = r393587 + r393588;
        double r393590 = x;
        double r393591 = r393590 / r393579;
        double r393592 = r393589 * r393591;
        double r393593 = r393592 - r393587;
        double r393594 = log(r393593);
        double r393595 = r393586 - r393594;
        double r393596 = r393586 - r393579;
        double r393597 = cbrt(r393596);
        double r393598 = r393597 * r393597;
        double r393599 = r393588 / r393598;
        double r393600 = r393590 - r393579;
        double r393601 = r393600 / r393597;
        double r393602 = r393599 * r393601;
        double r393603 = r393586 - r393602;
        double r393604 = log(r393603);
        double r393605 = r393586 - r393604;
        double r393606 = r393585 ? r393595 : r393605;
        return r393606;
}

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))))))