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

Average Error: 18.8 → 0.2

Time: 22.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r226109 = 1.0;
        double r226110 = x;
        double r226111 = y;
        double r226112 = r226110 - r226111;
        double r226113 = r226109 - r226111;
        double r226114 = r226112 / r226113;
        double r226115 = r226109 - r226114;
        double r226116 = log(r226115);
        double r226117 = r226109 - r226116;
        return r226117;
}

↓

double f(double x, double y) {
        double r226118 = x;
        double r226119 = y;
        double r226120 = r226118 - r226119;
        double r226121 = 1.0;
        double r226122 = r226121 - r226119;
        double r226123 = r226120 / r226122;
        double r226124 = 0.967940061387686;
        bool r226125 = r226123 <= r226124;
        double r226126 = exp(r226121);
        double r226127 = r226121 - r226123;
        double r226128 = r226126 / r226127;
        double r226129 = log(r226128);
        double r226130 = 2.0;
        double r226131 = pow(r226119, r226130);
        double r226132 = r226118 / r226131;
        double r226133 = r226118 / r226119;
        double r226134 = fma(r226121, r226132, r226133);
        double r226135 = r226121 / r226119;
        double r226136 = r226134 - r226135;
        double r226137 = cbrt(r226122);
        double r226138 = 3.0;
        double r226139 = pow(r226137, r226138);
        double r226140 = r226120 / r226139;
        double r226141 = -r226140;
        double r226142 = r226140 + r226141;
        double r226143 = r226136 + r226142;
        double r226144 = log(r226143);
        double r226145 = r226121 - r226144;
        double r226146 = r226125 ? r226129 : r226145;
        return r226146;
}

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-sqr-sqrt 62.1

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

   7. Applied prod-diff 62.1

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

   8. Simplified 57.1

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

   9. Simplified 57.1

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

   10. Taylor expanded around inf 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

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 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))))))