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

Average Error: 18.3 → 0.1

Time: 4.6s

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 r492317 = 1.0;
        double r492318 = x;
        double r492319 = y;
        double r492320 = r492318 - r492319;
        double r492321 = r492317 - r492319;
        double r492322 = r492320 / r492321;
        double r492323 = r492317 - r492322;
        double r492324 = log(r492323);
        double r492325 = r492317 - r492324;
        return r492325;
}

↓

double f(double x, double y) {
        double r492326 = x;
        double r492327 = y;
        double r492328 = r492326 - r492327;
        double r492329 = 1.0;
        double r492330 = r492329 - r492327;
        double r492331 = r492328 / r492330;
        double r492332 = 0.9987383373401453;
        bool r492333 = r492331 <= r492332;
        double r492334 = 1.0;
        double r492335 = r492334 / r492330;
        double r492336 = r492328 * r492335;
        double r492337 = r492329 - r492336;
        double r492338 = log(r492337);
        double r492339 = r492329 - r492338;
        double r492340 = 2.0;
        double r492341 = pow(r492327, r492340);
        double r492342 = r492326 / r492341;
        double r492343 = r492334 / r492327;
        double r492344 = r492342 - r492343;
        double r492345 = r492329 * r492344;
        double r492346 = r492326 / r492327;
        double r492347 = r492345 + r492346;
        double r492348 = log(r492347);
        double r492349 = r492329 - r492348;
        double r492350 = r492333 ? r492339 : r492349;
        return r492350;
}

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))))))