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

Average Error: 18.1 → 0.2

Time: 11.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -78185572.3768116533756256103515625 \lor \neg \left(y \le 0.9979648952928983351995384509791620075703\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\mathsf{fma}\left(1, 1, -\frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}\right)}\right) + \log \left(\sqrt{1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r359231 = 1.0;
        double r359232 = x;
        double r359233 = y;
        double r359234 = r359232 - r359233;
        double r359235 = r359231 - r359233;
        double r359236 = r359234 / r359235;
        double r359237 = r359231 - r359236;
        double r359238 = log(r359237);
        double r359239 = r359231 - r359238;
        return r359239;
}

↓

double f(double x, double y) {
        double r359240 = y;
        double r359241 = -78185572.37681165;
        bool r359242 = r359240 <= r359241;
        double r359243 = 0.9979648952928983;
        bool r359244 = r359240 <= r359243;
        double r359245 = !r359244;
        bool r359246 = r359242 || r359245;
        double r359247 = 1.0;
        double r359248 = x;
        double r359249 = r359248 / r359240;
        double r359250 = 2.0;
        double r359251 = pow(r359240, r359250);
        double r359252 = r359248 / r359251;
        double r359253 = r359247 * r359252;
        double r359254 = r359249 + r359253;
        double r359255 = 1.0;
        double r359256 = r359255 / r359240;
        double r359257 = r359247 * r359256;
        double r359258 = r359254 - r359257;
        double r359259 = log(r359258);
        double r359260 = r359247 - r359259;
        double r359261 = r359247 * r359247;
        double r359262 = r359240 * r359240;
        double r359263 = r359247 * r359240;
        double r359264 = r359262 + r359263;
        double r359265 = r359261 + r359264;
        double r359266 = cbrt(r359265);
        double r359267 = r359266 * r359266;
        double r359268 = r359247 - r359240;
        double r359269 = cbrt(r359268);
        double r359270 = sqrt(r359269);
        double r359271 = r359267 / r359270;
        double r359272 = r359248 - r359240;
        double r359273 = 3.0;
        double r359274 = pow(r359247, r359273);
        double r359275 = pow(r359240, r359273);
        double r359276 = r359274 - r359275;
        double r359277 = cbrt(r359276);
        double r359278 = r359277 * r359277;
        double r359279 = r359272 / r359278;
        double r359280 = r359279 / r359270;
        double r359281 = r359271 * r359280;
        double r359282 = -r359281;
        double r359283 = fma(r359255, r359247, r359282);
        double r359284 = -r359271;
        double r359285 = fma(r359284, r359280, r359281);
        double r359286 = r359283 + r359285;
        double r359287 = sqrt(r359286);
        double r359288 = log(r359287);
        double r359289 = r359269 * r359269;
        double r359290 = r359272 / r359289;
        double r359291 = r359290 / r359269;
        double r359292 = r359247 - r359291;
        double r359293 = sqrt(r359292);
        double r359294 = log(r359293);
        double r359295 = r359288 + r359294;
        double r359296 = r359247 - r359295;
        double r359297 = r359246 ? r359260 : r359296;
        return r359297;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.1
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if y < -78185572.37681165 or 0.9979648952928983 < y

   1. Initial program 46.2

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

   3. Applied add-cube-cbrt 42.5

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

   4. Applied associate-/r* 42.5

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

   5. Taylor expanded around inf 0.2

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

   if -78185572.37681165 < y < 0.9979648952928983

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   4. Applied associate-/r* 0.1

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

   6. Applied add-sqr-sqrt 0.2

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

   7. Applied log-prod 0.1

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\right)}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 0.1

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

   10. Applied flip3-- 0.1

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

   11. Applied cbrt-div 0.1

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

   12. Applied flip3-- 0.1

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

   13. Applied cbrt-div 0.1

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

   14. Applied frac-times 0.1

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

   15. Applied associate-/r/ 0.1

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

   16. Applied times-frac 0.1

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

   17. Applied add-sqr-sqrt 0.1

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

   18. Applied prod-diff 0.1

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

   19. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -78185572.3768116533756256103515625 \lor \neg \left(y \le 0.9979648952928983351995384509791620075703\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\mathsf{fma}\left(1, 1, -\frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)} \cdot \sqrt[3]{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{{1}^{3} - {y}^{3}} \cdot \sqrt[3]{{1}^{3} - {y}^{3}}}}{\sqrt{\sqrt[3]{1 - y}}}\right)}\right) + \log \left(\sqrt{1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 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))))))