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

Average Error: 18.3 → 0.1

Time: 28.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, \frac{\frac{-1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(\frac{\frac{-1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}}, \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}, \frac{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r13884120 = 1.0;
        double r13884121 = x;
        double r13884122 = y;
        double r13884123 = r13884121 - r13884122;
        double r13884124 = r13884120 - r13884122;
        double r13884125 = r13884123 / r13884124;
        double r13884126 = r13884120 - r13884125;
        double r13884127 = log(r13884126);
        double r13884128 = r13884120 - r13884127;
        return r13884128;
}

↓

double f(double x, double y) {
        double r13884129 = y;
        double r13884130 = -60638491.817888685;
        bool r13884131 = r13884129 <= r13884130;
        double r13884132 = 1.0;
        double r13884133 = x;
        double r13884134 = r13884133 / r13884129;
        double r13884135 = r13884132 / r13884129;
        double r13884136 = r13884134 - r13884135;
        double r13884137 = fma(r13884134, r13884135, r13884136);
        double r13884138 = log(r13884137);
        double r13884139 = r13884132 - r13884138;
        double r13884140 = 4.2398846312504237e+33;
        bool r13884141 = r13884129 <= r13884140;
        double r13884142 = 1.0;
        double r13884143 = -1.0;
        double r13884144 = r13884132 - r13884129;
        double r13884145 = cbrt(r13884144);
        double r13884146 = r13884145 * r13884145;
        double r13884147 = r13884143 / r13884146;
        double r13884148 = cbrt(r13884145);
        double r13884149 = r13884147 / r13884148;
        double r13884150 = r13884133 - r13884129;
        double r13884151 = cbrt(r13884146);
        double r13884152 = r13884150 / r13884151;
        double r13884153 = r13884149 * r13884152;
        double r13884154 = fma(r13884142, r13884132, r13884153);
        double r13884155 = r13884142 / r13884146;
        double r13884156 = r13884155 / r13884148;
        double r13884157 = r13884156 * r13884152;
        double r13884158 = fma(r13884149, r13884152, r13884157);
        double r13884159 = r13884154 + r13884158;
        double r13884160 = log(r13884159);
        double r13884161 = r13884132 - r13884160;
        double r13884162 = r13884141 ? r13884161 : r13884139;
        double r13884163 = r13884131 ? r13884139 : r13884162;
        return r13884163;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -60638491.817888685 or 4.2398846312504237e+33 < y

   1. Initial program 47.4

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

   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

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

   if -60638491.817888685 < y < 4.2398846312504237e+33

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   4. Applied associate-/r* 0.1

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

   6. Applied add-cube-cbrt 0.1

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

   7. Applied cbrt-prod 0.1

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

   8. Applied div-inv 0.1

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

   9. Applied times-frac 0.1

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

   10. Applied *-un-lft-identity 0.1

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

   11. Applied prod-diff 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, \frac{\frac{-1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(\frac{\frac{-1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}}, \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}, \frac{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))