Average Error: 0.2 → 0.2
Time: 38.0s
Precision: 64
\[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(\frac{\frac{-1}{3.0}}{\sqrt{x}}, y, y \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}\right) + \mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, y \cdot \frac{\frac{-1}{3.0}}{\sqrt{x}}\right)\]
\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}
↓
\mathsf{fma}\left(\frac{\frac{-1}{3.0}}{\sqrt{x}}, y, y \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}\right) + \mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, y \cdot \frac{\frac{-1}{3.0}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r17893782 = 1.0;
        double r17893783 = x;
        double r17893784 = 9.0;
        double r17893785 = r17893783 * r17893784;
        double r17893786 = r17893782 / r17893785;
        double r17893787 = r17893782 - r17893786;
        double r17893788 = y;
        double r17893789 = 3.0;
        double r17893790 = sqrt(r17893783);
        double r17893791 = r17893789 * r17893790;
        double r17893792 = r17893788 / r17893791;
        double r17893793 = r17893787 - r17893792;
        return r17893793;
}
↓
double f(double x, double y) {
        double r17893794 = -1.0;
        double r17893795 = 3.0;
        double r17893796 = r17893794 / r17893795;
        double r17893797 = x;
        double r17893798 = sqrt(r17893797);
        double r17893799 = r17893796 / r17893798;
        double r17893800 = y;
        double r17893801 = 1.0;
        double r17893802 = r17893801 / r17893795;
        double r17893803 = r17893802 / r17893798;
        double r17893804 = r17893800 * r17893803;
        double r17893805 = fma(r17893799, r17893800, r17893804);
        double r17893806 = 1.0;
        double r17893807 = 9.0;
        double r17893808 = r17893797 * r17893807;
        double r17893809 = r17893806 / r17893808;
        double r17893810 = r17893806 - r17893809;
        double r17893811 = r17893800 * r17893799;
        double r17893812 = fma(r17893801, r17893810, r17893811);
        double r17893813 = r17893805 + r17893812;
        return r17893813;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[\left(1.0 - \frac{\frac{1.0}{x}}{9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{y}{3.0 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{\frac{y}{3.0}}{\sqrt{x}}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\frac{y}{3.0}}{\color{blue}{1 \cdot \sqrt{x}}}\]

  6. Applied div-inv0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \frac{\color{blue}{y \cdot \frac{1}{3.0}}}{1 \cdot \sqrt{x}}\]

  7. Applied times-frac0.2

    \[\leadsto \left(1.0 - \frac{1.0}{x \cdot 9.0}\right) - \color{blue}{\frac{y}{1} \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}}\]

  8. Applied *-un-lft-identity0.2

    \[\leadsto \color{blue}{1 \cdot \left(1.0 - \frac{1.0}{x \cdot 9.0}\right)} - \frac{y}{1} \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}\]

  9. Applied prod-diff0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, -\frac{\frac{1}{3.0}}{\sqrt{x}} \cdot \frac{y}{1}\right) + \mathsf{fma}\left(-\frac{\frac{1}{3.0}}{\sqrt{x}}, \frac{y}{1}, \frac{\frac{1}{3.0}}{\sqrt{x}} \cdot \frac{y}{1}\right)}\]

  10. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{3.0}}{\sqrt{x}}, y, y \cdot \frac{\frac{1}{3.0}}{\sqrt{x}}\right) + \mathsf{fma}\left(1, 1.0 - \frac{1.0}{x \cdot 9.0}, y \cdot \frac{\frac{-1}{3.0}}{\sqrt{x}}\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))