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

Average Error: 18.0 → 0.2

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999975065411717:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r398646 = 1.0;
        double r398647 = x;
        double r398648 = y;
        double r398649 = r398647 - r398648;
        double r398650 = r398646 - r398648;
        double r398651 = r398649 / r398650;
        double r398652 = r398646 - r398651;
        double r398653 = log(r398652);
        double r398654 = r398646 - r398653;
        return r398654;
}

↓

double f(double x, double y) {
        double r398655 = x;
        double r398656 = y;
        double r398657 = r398655 - r398656;
        double r398658 = 1.0;
        double r398659 = r398658 - r398656;
        double r398660 = r398657 / r398659;
        double r398661 = 0.9999997506541172;
        bool r398662 = r398660 <= r398661;
        double r398663 = cbrt(r398657);
        double r398664 = r398663 * r398663;
        double r398665 = r398663 / r398659;
        double r398666 = r398664 * r398665;
        double r398667 = r398658 - r398666;
        double r398668 = log(r398667);
        double r398669 = r398658 - r398668;
        double r398670 = 2.0;
        double r398671 = pow(r398656, r398670);
        double r398672 = r398655 / r398671;
        double r398673 = 1.0;
        double r398674 = r398673 / r398656;
        double r398675 = r398672 - r398674;
        double r398676 = r398658 * r398675;
        double r398677 = r398655 / r398656;
        double r398678 = r398676 + r398677;
        double r398679 = log(r398678);
        double r398680 = r398658 - r398679;
        double r398681 = r398662 ? r398669 : r398680;
        return r398681;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.0
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 (/ (- x y) (- 1.0 y)) < 0.9999997506541172

   1. Initial program 0.1

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

   3. Applied *-un-lft-identity 0.1

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

   4. Applied add-cube-cbrt 0.1

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

   5. Applied times-frac 0.1

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

   6. Simplified 0.1

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

   if 0.9999997506541172 < (/ (- x y) (- 1.0 y))

   1. Initial program 62.3

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999975065411717:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))))))