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

Average Error: 18.5 → 0.1

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999999374492958393:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r455168 = 1.0;
        double r455169 = x;
        double r455170 = y;
        double r455171 = r455169 - r455170;
        double r455172 = r455168 - r455170;
        double r455173 = r455171 / r455172;
        double r455174 = r455168 - r455173;
        double r455175 = log(r455174);
        double r455176 = r455168 - r455175;
        return r455176;
}

↓

double f(double x, double y) {
        double r455177 = x;
        double r455178 = y;
        double r455179 = r455177 - r455178;
        double r455180 = 1.0;
        double r455181 = r455180 - r455178;
        double r455182 = r455179 / r455181;
        double r455183 = 0.9999993744929584;
        bool r455184 = r455182 <= r455183;
        double r455185 = 1.0;
        double r455186 = r455185 / r455181;
        double r455187 = r455179 * r455186;
        double r455188 = r455180 - r455187;
        double r455189 = log(r455188);
        double r455190 = r455180 - r455189;
        double r455191 = 2.0;
        double r455192 = pow(r455178, r455191);
        double r455193 = r455177 / r455192;
        double r455194 = r455185 / r455178;
        double r455195 = r455193 - r455194;
        double r455196 = r455180 * r455195;
        double r455197 = r455177 / r455178;
        double r455198 = r455196 + r455197;
        double r455199 = log(r455198);
        double r455200 = r455180 - r455199;
        double r455201 = r455184 ? r455190 : r455200;
        return r455201;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.5
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 0.9999993744929584

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

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

   if 0.9999993744929584 < (/ (- x y) (- 1.0 y))

   1. Initial program 62.5

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

   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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.999999374492958393:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))))))