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

Average Error: 18.5 → 0.1

Time: 21.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right) + \mathsf{fma}\left(1, 1, \left(1 + y\right) \cdot \left(-\frac{x - y}{1 \cdot 1 - y \cdot y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r15640083 = 1.0;
        double r15640084 = x;
        double r15640085 = y;
        double r15640086 = r15640084 - r15640085;
        double r15640087 = r15640083 - r15640085;
        double r15640088 = r15640086 / r15640087;
        double r15640089 = r15640083 - r15640088;
        double r15640090 = log(r15640089);
        double r15640091 = r15640083 - r15640090;
        return r15640091;
}

↓

double f(double x, double y) {
        double r15640092 = y;
        double r15640093 = -34474611802046.883;
        bool r15640094 = r15640092 <= r15640093;
        double r15640095 = 1.0;
        double r15640096 = r15640095 / r15640092;
        double r15640097 = x;
        double r15640098 = r15640097 / r15640092;
        double r15640099 = r15640098 - r15640096;
        double r15640100 = fma(r15640096, r15640098, r15640099);
        double r15640101 = log(r15640100);
        double r15640102 = r15640095 - r15640101;
        double r15640103 = 43744445.70071104;
        bool r15640104 = r15640092 <= r15640103;
        double r15640105 = r15640095 + r15640092;
        double r15640106 = -r15640105;
        double r15640107 = r15640097 - r15640092;
        double r15640108 = r15640095 * r15640095;
        double r15640109 = r15640092 * r15640092;
        double r15640110 = r15640108 - r15640109;
        double r15640111 = r15640107 / r15640110;
        double r15640112 = r15640105 * r15640111;
        double r15640113 = fma(r15640106, r15640111, r15640112);
        double r15640114 = 1.0;
        double r15640115 = -r15640111;
        double r15640116 = r15640105 * r15640115;
        double r15640117 = fma(r15640114, r15640095, r15640116);
        double r15640118 = r15640113 + r15640117;
        double r15640119 = log(r15640118);
        double r15640120 = r15640095 - r15640119;
        double r15640121 = r15640104 ? r15640120 : r15640102;
        double r15640122 = r15640094 ? r15640102 : r15640121;
        return r15640122;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.5
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 < -34474611802046.883 or 43744445.70071104 < y

   1. Initial program 47.4

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -34474611802046.883 < y < 43744445.70071104

   1. Initial program 0.2

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

   3. Applied flip-- 0.2

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

   4. Applied associate-/r/ 0.2

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

   5. Applied *-un-lft-identity 0.2

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

   6. Applied prod-diff 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right) + \mathsf{fma}\left(1, 1, \left(1 + y\right) \cdot \left(-\frac{x - y}{1 \cdot 1 - y \cdot y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))))