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

Average Error: 18.3 → 0.4

Time: 25.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.004440397313824472058652048644944443367422:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r261106 = 1.0;
        double r261107 = x;
        double r261108 = y;
        double r261109 = r261107 - r261108;
        double r261110 = r261106 - r261108;
        double r261111 = r261109 / r261110;
        double r261112 = r261106 - r261111;
        double r261113 = log(r261112);
        double r261114 = r261106 - r261113;
        return r261114;
}

↓

double f(double x, double y) {
        double r261115 = x;
        double r261116 = y;
        double r261117 = r261115 - r261116;
        double r261118 = 1.0;
        double r261119 = r261118 - r261116;
        double r261120 = r261117 / r261119;
        double r261121 = 0.004440397313824472;
        bool r261122 = r261120 <= r261121;
        double r261123 = cbrt(r261119);
        double r261124 = 3.0;
        double r261125 = pow(r261123, r261124);
        double r261126 = r261117 / r261125;
        double r261127 = r261118 - r261126;
        double r261128 = -r261126;
        double r261129 = r261126 + r261128;
        double r261130 = r261127 + r261129;
        double r261131 = log(r261130);
        double r261132 = r261118 - r261131;
        double r261133 = 2.0;
        double r261134 = pow(r261116, r261133);
        double r261135 = r261115 / r261134;
        double r261136 = r261115 / r261116;
        double r261137 = fma(r261118, r261135, r261136);
        double r261138 = r261118 / r261116;
        double r261139 = r261137 - r261138;
        double r261140 = r261139 + r261129;
        double r261141 = log(r261140);
        double r261142 = r261118 - r261141;
        double r261143 = r261122 ? r261132 : r261142;
        return r261143;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
Target	0.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 0.004440397313824472

   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-sqr-sqrt 0.0

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

   7. Applied prod-diff 0.0

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

   8. Simplified 0.0

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

   9. Simplified 0.0

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

   if 0.004440397313824472 < (/ (- x y) (- 1.0 y))

   1. Initial program 60.2

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

   3. Applied add-cube-cbrt 55.4

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

   4. Applied add-cube-cbrt 60.4

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

   5. Applied times-frac 60.4

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

   6. Applied add-sqr-sqrt 60.4

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

   7. Applied prod-diff 60.4

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

   8. Simplified 55.5

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

   9. Simplified 55.4

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

   10. Taylor expanded around inf 1.4

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

   11. Simplified 1.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.004440397313824472058652048644944443367422:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))))