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

Average Error: 18.8 → 0.2

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9679400613876859571504951418319251388311:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r303130 = 1.0;
        double r303131 = x;
        double r303132 = y;
        double r303133 = r303131 - r303132;
        double r303134 = r303130 - r303132;
        double r303135 = r303133 / r303134;
        double r303136 = r303130 - r303135;
        double r303137 = log(r303136);
        double r303138 = r303130 - r303137;
        return r303138;
}

↓

double f(double x, double y) {
        double r303139 = x;
        double r303140 = y;
        double r303141 = r303139 - r303140;
        double r303142 = 1.0;
        double r303143 = r303142 - r303140;
        double r303144 = r303141 / r303143;
        double r303145 = 0.967940061387686;
        bool r303146 = r303144 <= r303145;
        double r303147 = exp(r303142);
        double r303148 = r303142 - r303144;
        double r303149 = r303147 / r303148;
        double r303150 = log(r303149);
        double r303151 = 2.0;
        double r303152 = pow(r303140, r303151);
        double r303153 = r303139 / r303152;
        double r303154 = r303139 / r303140;
        double r303155 = fma(r303153, r303142, r303154);
        double r303156 = r303142 / r303140;
        double r303157 = r303155 - r303156;
        double r303158 = cbrt(r303143);
        double r303159 = 3.0;
        double r303160 = pow(r303158, r303159);
        double r303161 = r303141 / r303160;
        double r303162 = -r303161;
        double r303163 = r303161 + r303162;
        double r303164 = r303157 + r303163;
        double r303165 = log(r303164);
        double r303166 = r303142 - r303165;
        double r303167 = r303146 ? r303150 : r303166;
        return r303167;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.8
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 (/ (- x y) (- 1.0 y)) < 0.967940061387686

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied diff-log 0.0

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

   if 0.967940061387686 < (/ (- x y) (- 1.0 y))

   1. Initial program 61.9

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

   3. Applied add-cube-cbrt 57.2

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

   4. Applied add-cube-cbrt 62.1

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

   5. Applied times-frac 62.1

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

   6. Applied add-cube-cbrt 62.1

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

   7. Applied prod-diff 62.1

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

   8. Simplified 57.1

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

   9. Simplified 57.1

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

   10. Taylor expanded around inf 0.5

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

   11. Simplified 0.5

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9679400613876859571504951418319251388311:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right) + \left(\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}} + \left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))))