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

Average Error: 18.0 → 0.1

Time: 7.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -142834896.66603637 \lor \neg \left(y \le 28334228.2028031461\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{y} \cdot \frac{\sqrt[3]{x}}{y} - {\left(\frac{\sqrt[3]{1}}{\sqrt[3]{y}}\right)}^{3}\right) + \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(-\frac{1}{\sqrt[3]{y}}\right) + \frac{1}{\sqrt[3]{y}}\right), \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r352876 = 1.0;
        double r352877 = x;
        double r352878 = y;
        double r352879 = r352877 - r352878;
        double r352880 = r352876 - r352878;
        double r352881 = r352879 / r352880;
        double r352882 = r352876 - r352881;
        double r352883 = log(r352882);
        double r352884 = r352876 - r352883;
        return r352884;
}

↓

double f(double x, double y) {
        double r352885 = y;
        double r352886 = -142834896.66603637;
        bool r352887 = r352885 <= r352886;
        double r352888 = 28334228.202803146;
        bool r352889 = r352885 <= r352888;
        double r352890 = !r352889;
        bool r352891 = r352887 || r352890;
        double r352892 = 1.0;
        double r352893 = x;
        double r352894 = cbrt(r352893);
        double r352895 = r352894 * r352894;
        double r352896 = r352895 / r352885;
        double r352897 = r352894 / r352885;
        double r352898 = r352896 * r352897;
        double r352899 = 1.0;
        double r352900 = cbrt(r352899);
        double r352901 = cbrt(r352885);
        double r352902 = r352900 / r352901;
        double r352903 = 3.0;
        double r352904 = pow(r352902, r352903);
        double r352905 = r352898 - r352904;
        double r352906 = r352901 * r352901;
        double r352907 = r352899 / r352906;
        double r352908 = r352899 / r352901;
        double r352909 = -r352908;
        double r352910 = r352909 + r352908;
        double r352911 = r352907 * r352910;
        double r352912 = r352905 + r352911;
        double r352913 = r352893 / r352885;
        double r352914 = fma(r352892, r352912, r352913);
        double r352915 = log(r352914);
        double r352916 = r352892 - r352915;
        double r352917 = r352892 - r352885;
        double r352918 = cbrt(r352917);
        double r352919 = r352918 * r352918;
        double r352920 = r352899 / r352919;
        double r352921 = r352893 - r352885;
        double r352922 = r352921 / r352918;
        double r352923 = r352920 * r352922;
        double r352924 = r352892 - r352923;
        double r352925 = log(r352924);
        double r352926 = r352892 - r352925;
        double r352927 = r352891 ? r352916 : r352926;
        return r352927;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.0
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 y < -142834896.66603637 or 28334228.202803146 < y

   1. Initial program 46.3

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.1

      \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}, \frac{x}{y}\right)\right)\]

   6. Applied add-cube-cbrt 0.1

      \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}, \frac{x}{y}\right)\right)\]

   7. Applied times-frac 0.1

      \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}}, \frac{x}{y}\right)\right)\]

   8. Applied add-sqr-sqrt 47.4

      \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}}^{2}} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}, \frac{x}{y}\right)\right)\]

   9. Applied unpow-prod-down 47.4

      \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{x}{\color{blue}{{\left(\sqrt{y}\right)}^{2} \cdot {\left(\sqrt{y}\right)}^{2}}} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}, \frac{x}{y}\right)\right)\]

   10. Applied add-cube-cbrt 47.4

       \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{{\left(\sqrt{y}\right)}^{2} \cdot {\left(\sqrt{y}\right)}^{2}} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}, \frac{x}{y}\right)\right)\]

   11. Applied times-frac 47.4

       \[\leadsto 1 - \log \left(\mathsf{fma}\left(1, \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{{\left(\sqrt{y}\right)}^{2}} \cdot \frac{\sqrt[3]{x}}{{\left(\sqrt{y}\right)}^{2}}} - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{y}}, \frac{x}{y}\right)\right)\]

   12. Applied prod-diff 47.4

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

   13. Simplified 0.1

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

   14. Simplified 0.1

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

   if -142834896.66603637 < y < 28334228.202803146

   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 *-un-lft-identity 0.1

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

   5. Applied times-frac 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -142834896.66603637 \lor \neg \left(y \le 28334228.2028031461\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{y} \cdot \frac{\sqrt[3]{x}}{y} - {\left(\frac{\sqrt[3]{1}}{\sqrt[3]{y}}\right)}^{3}\right) + \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(-\frac{1}{\sqrt[3]{y}}\right) + \frac{1}{\sqrt[3]{y}}\right), \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))))