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

Average Error: 17.8 → 0.5

Time: 25.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 9.496174125024586353937400273750313317578 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r296307 = 1.0;
        double r296308 = x;
        double r296309 = y;
        double r296310 = r296308 - r296309;
        double r296311 = r296307 - r296309;
        double r296312 = r296310 / r296311;
        double r296313 = r296307 - r296312;
        double r296314 = log(r296313);
        double r296315 = r296307 - r296314;
        return r296315;
}

↓

double f(double x, double y) {
        double r296316 = x;
        double r296317 = y;
        double r296318 = r296316 - r296317;
        double r296319 = 1.0;
        double r296320 = r296319 - r296317;
        double r296321 = r296318 / r296320;
        double r296322 = 9.496174125024586e-05;
        bool r296323 = r296321 <= r296322;
        double r296324 = cbrt(r296319);
        double r296325 = 3.0;
        double r296326 = pow(r296324, r296325);
        double r296327 = cbrt(r296320);
        double r296328 = pow(r296327, r296325);
        double r296329 = r296318 / r296328;
        double r296330 = r296326 - r296329;
        double r296331 = -r296329;
        double r296332 = r296329 + r296331;
        double r296333 = r296330 + r296332;
        double r296334 = log(r296333);
        double r296335 = r296319 - r296334;
        double r296336 = 2.0;
        double r296337 = pow(r296317, r296336);
        double r296338 = r296316 / r296337;
        double r296339 = r296316 / r296317;
        double r296340 = fma(r296338, r296319, r296339);
        double r296341 = r296319 / r296317;
        double r296342 = r296340 - r296341;
        double r296343 = log(r296342);
        double r296344 = r296319 - r296343;
        double r296345 = r296323 ? r296335 : r296344;
        return r296345;
}

Error

Bits error versus x

Bits error versus y

Target

Original	17.8
Target	0.1
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 9.496174125024586e-05

   1. Initial program 0.0

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

   3. Applied add-cube-cbrt 0.0

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

   4. Applied add-cube-cbrt 0.0

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

   5. Applied times-frac 0.0

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

   6. Applied add-cube-cbrt 0.0

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

   7. Applied prod-diff 0.0

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

   8. Simplified 0.0

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

   9. Simplified 0.0

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

   if 9.496174125024586e-05 < (/ (- x y) (- 1.0 y))

   1. Initial program 59.8

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

   2. Taylor expanded around inf 1.6

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

   3. Simplified 1.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 9.496174125024586353937400273750313317578 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))