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

Average Error: 18.3 → 1.2

Time: 20.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -9.186493886184318 \cdot 10^{36} \lor \neg \left(y \le 49288711.7499881461\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r330261 = 1.0;
        double r330262 = x;
        double r330263 = y;
        double r330264 = r330262 - r330263;
        double r330265 = r330261 - r330263;
        double r330266 = r330264 / r330265;
        double r330267 = r330261 - r330266;
        double r330268 = log(r330267);
        double r330269 = r330261 - r330268;
        return r330269;
}

↓

double f(double x, double y) {
        double r330270 = y;
        double r330271 = -9.186493886184318e+36;
        bool r330272 = r330270 <= r330271;
        double r330273 = 49288711.749988146;
        bool r330274 = r330270 <= r330273;
        double r330275 = !r330274;
        bool r330276 = r330272 || r330275;
        double r330277 = 1.0;
        double r330278 = x;
        double r330279 = r330278 / r330270;
        double r330280 = 2.0;
        double r330281 = pow(r330270, r330280);
        double r330282 = r330278 / r330281;
        double r330283 = r330277 * r330282;
        double r330284 = r330279 + r330283;
        double r330285 = r330277 / r330270;
        double r330286 = r330284 - r330285;
        double r330287 = log(r330286);
        double r330288 = r330277 - r330287;
        double r330289 = r330278 - r330270;
        double r330290 = cbrt(r330289);
        double r330291 = r330290 * r330290;
        double r330292 = r330277 - r330270;
        double r330293 = r330290 / r330292;
        double r330294 = r330291 * r330293;
        double r330295 = r330277 - r330294;
        double r330296 = log(r330295);
        double r330297 = r330277 - r330296;
        double r330298 = r330276 ? r330288 : r330297;
        return r330298;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
Target	0.1
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if y < -9.186493886184318e+36 or 49288711.749988146 < y

   1. Initial program 48.0

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -9.186493886184318e+36 < y < 49288711.749988146

   1. Initial program 2.0

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

   3. Applied *-un-lft-identity 2.0

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

   4. Applied add-cube-cbrt 1.9

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

   5. Applied times-frac 1.9

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

   6. Simplified 1.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.186493886184318 \cdot 10^{36} \lor \neg \left(y \le 49288711.7499881461\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))