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

Average Error: 18.3 → 0.2

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 2.61378141 \cdot 10^{-10}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r422787 = 1.0;
        double r422788 = x;
        double r422789 = y;
        double r422790 = r422788 - r422789;
        double r422791 = r422787 - r422789;
        double r422792 = r422790 / r422791;
        double r422793 = r422787 - r422792;
        double r422794 = log(r422793);
        double r422795 = r422787 - r422794;
        return r422795;
}

↓

double f(double x, double y) {
        double r422796 = 1.0;
        double r422797 = x;
        double r422798 = y;
        double r422799 = r422797 - r422798;
        double r422800 = r422796 - r422798;
        double r422801 = r422799 / r422800;
        double r422802 = r422796 - r422801;
        double r422803 = 2.6137814135296367e-10;
        bool r422804 = r422802 <= r422803;
        double r422805 = 2.0;
        double r422806 = pow(r422798, r422805);
        double r422807 = r422797 / r422806;
        double r422808 = 1.0;
        double r422809 = r422808 / r422798;
        double r422810 = r422807 - r422809;
        double r422811 = r422796 * r422810;
        double r422812 = r422797 / r422798;
        double r422813 = r422811 + r422812;
        double r422814 = log(r422813);
        double r422815 = r422796 - r422814;
        double r422816 = cbrt(r422799);
        double r422817 = r422816 * r422816;
        double r422818 = r422816 / r422800;
        double r422819 = r422817 * r422818;
        double r422820 = r422796 - r422819;
        double r422821 = log(r422820);
        double r422822 = r422796 - r422821;
        double r422823 = r422804 ? r422815 : r422822;
        return r422823;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
Target	0.1
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- 1.0 (/ (- x y) (- 1.0 y))) < 2.6137814135296367e-10

   1. Initial program 63.4

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

   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

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

   if 2.6137814135296367e-10 < (- 1.0 (/ (- x y) (- 1.0 y)))

   1. Initial program 0.2

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

   3. Applied *-un-lft-identity 0.2

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

   4. Applied add-cube-cbrt 0.2

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

   5. Applied times-frac 0.3

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

   6. Simplified 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \le 2.61378141 \cdot 10^{-10}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

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