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

Average Error: 18.7 → 0.3

Time: 35.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.07944260022352443717519321353393024764955:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - 1 \cdot \frac{1}{y}\right) + \left(\left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r268649 = 1.0;
        double r268650 = x;
        double r268651 = y;
        double r268652 = r268650 - r268651;
        double r268653 = r268649 - r268651;
        double r268654 = r268652 / r268653;
        double r268655 = r268649 - r268654;
        double r268656 = log(r268655);
        double r268657 = r268649 - r268656;
        return r268657;
}

↓

double f(double x, double y) {
        double r268658 = x;
        double r268659 = y;
        double r268660 = r268658 - r268659;
        double r268661 = 1.0;
        double r268662 = r268661 - r268659;
        double r268663 = r268660 / r268662;
        double r268664 = 0.07944260022352444;
        bool r268665 = r268663 <= r268664;
        double r268666 = cbrt(r268661);
        double r268667 = 3.0;
        double r268668 = pow(r268666, r268667);
        double r268669 = cbrt(r268662);
        double r268670 = pow(r268669, r268667);
        double r268671 = r268660 / r268670;
        double r268672 = r268668 - r268671;
        double r268673 = -r268671;
        double r268674 = r268673 + r268671;
        double r268675 = r268672 + r268674;
        double r268676 = log(r268675);
        double r268677 = r268661 - r268676;
        double r268678 = 2.0;
        double r268679 = pow(r268659, r268678);
        double r268680 = r268658 / r268679;
        double r268681 = r268658 / r268659;
        double r268682 = fma(r268680, r268661, r268681);
        double r268683 = 1.0;
        double r268684 = r268683 / r268659;
        double r268685 = r268661 * r268684;
        double r268686 = r268682 - r268685;
        double r268687 = r268686 + r268674;
        double r268688 = log(r268687);
        double r268689 = r268661 - r268688;
        double r268690 = r268665 ? r268677 : r268689;
        return r268690;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.7
Target	0.1
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 0.07944260022352444

   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-cube-cbrt 0.0

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

   7. Applied prod-diff 0.0

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

   8. Simplified 0.0

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

   9. Simplified 0.0

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

   if 0.07944260022352444 < (/ (- x y) (- 1.0 y))

   1. Initial program 61.4

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

   3. Applied add-cube-cbrt 56.4

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

   4. Applied add-cube-cbrt 61.6

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

   5. Applied times-frac 61.6

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

   6. Applied add-cube-cbrt 61.6

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

   7. Applied prod-diff 61.6

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

   8. Simplified 56.4

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

   9. Simplified 56.4

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

   10. Taylor expanded around inf 0.9

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

   11. Simplified 0.9

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.07944260022352443717519321353393024764955:\\ \;\;\;\;1 - \log \left(\left({\left(\sqrt[3]{1}\right)}^{3} - \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \left(\left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - 1 \cdot \frac{1}{y}\right) + \left(\left(-\frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right) + \frac{x - y}{{\left(\sqrt[3]{1 - y}\right)}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))