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

Average Error: 19.1 → 0.1

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -217320262.813254684 \lor \neg \left(y \le 298481706437129670000\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r372728 = 1.0;
        double r372729 = x;
        double r372730 = y;
        double r372731 = r372729 - r372730;
        double r372732 = r372728 - r372730;
        double r372733 = r372731 / r372732;
        double r372734 = r372728 - r372733;
        double r372735 = log(r372734);
        double r372736 = r372728 - r372735;
        return r372736;
}

↓

double f(double x, double y) {
        double r372737 = y;
        double r372738 = -217320262.81325468;
        bool r372739 = r372737 <= r372738;
        double r372740 = 2.9848170643712967e+20;
        bool r372741 = r372737 <= r372740;
        double r372742 = !r372741;
        bool r372743 = r372739 || r372742;
        double r372744 = 1.0;
        double r372745 = x;
        double r372746 = 2.0;
        double r372747 = pow(r372737, r372746);
        double r372748 = r372745 / r372747;
        double r372749 = 1.0;
        double r372750 = r372749 / r372737;
        double r372751 = r372748 - r372750;
        double r372752 = r372745 / r372737;
        double r372753 = fma(r372744, r372751, r372752);
        double r372754 = log(r372753);
        double r372755 = r372744 - r372754;
        double r372756 = r372745 - r372737;
        double r372757 = r372744 - r372737;
        double r372758 = r372749 / r372757;
        double r372759 = r372756 * r372758;
        double r372760 = r372744 - r372759;
        double r372761 = sqrt(r372760);
        double r372762 = log(r372761);
        double r372763 = expm1(r372762);
        double r372764 = log1p(r372763);
        double r372765 = r372764 + r372764;
        double r372766 = r372744 - r372765;
        double r372767 = r372743 ? r372755 : r372766;
        return r372767;
}

Error

Bits error versus x

Bits error versus y

Target

Original	19.1
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 < -217320262.81325468 or 2.9848170643712967e+20 < y

   1. Initial program 48.0

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

   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

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

   if -217320262.81325468 < y < 2.9848170643712967e+20

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

      \[\leadsto 1 - \log \left(1 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1 - y}}\right)\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied log-prod 0.1

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right) + \log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)}\]
   7. Using strategy rm

   8. Applied log1p-expm1-u 0.1

      \[\leadsto 1 - \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right)} + \log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\]
   9. Using strategy rm

   10. Applied log1p-expm1-u 0.1

       \[\leadsto 1 - \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right) + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -217320262.813254684 \lor \neg \left(y \le 298481706437129670000\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right) + \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{1 - \left(x - y\right) \cdot \frac{1}{1 - y}}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))))))