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

Average Error: 18.3 → 0.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99873833734014528:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r401528 = 1.0;
        double r401529 = x;
        double r401530 = y;
        double r401531 = r401529 - r401530;
        double r401532 = r401528 - r401530;
        double r401533 = r401531 / r401532;
        double r401534 = r401528 - r401533;
        double r401535 = log(r401534);
        double r401536 = r401528 - r401535;
        return r401536;
}

↓

double f(double x, double y) {
        double r401537 = x;
        double r401538 = y;
        double r401539 = r401537 - r401538;
        double r401540 = 1.0;
        double r401541 = r401540 - r401538;
        double r401542 = r401539 / r401541;
        double r401543 = 0.9987383373401453;
        bool r401544 = r401542 <= r401543;
        double r401545 = 1.0;
        double r401546 = r401545 / r401541;
        double r401547 = r401539 * r401546;
        double r401548 = r401540 - r401547;
        double r401549 = log(r401548);
        double r401550 = r401540 - r401549;
        double r401551 = 2.0;
        double r401552 = pow(r401538, r401551);
        double r401553 = r401537 / r401552;
        double r401554 = r401545 / r401538;
        double r401555 = r401553 - r401554;
        double r401556 = r401540 * r401555;
        double r401557 = r401537 / r401538;
        double r401558 = r401556 + r401557;
        double r401559 = log(r401558);
        double r401560 = r401540 - r401559;
        double r401561 = r401544 ? r401550 : r401560;
        return r401561;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
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)) < 0.9987383373401453

   1. Initial program 0.0

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

   3. Applied div-inv 0.0

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

   if 0.9987383373401453 < (/ (- x y) (- 1.0 y))

   1. Initial program 61.9

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99873833734014528:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))))