Average Error: 0.2 → 0.3
Time: 10.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \left(\frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \left(\frac{-1}{\sqrt[3]{9} \cdot x} + \frac{\frac{1}{x}}{\sqrt[3]{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \left(\frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \left(\frac{-1}{\sqrt[3]{9} \cdot x} + \frac{\frac{1}{x}}{\sqrt[3]{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r382749 = 1.0;
        double r382750 = x;
        double r382751 = 9.0;
        double r382752 = r382750 * r382751;
        double r382753 = r382749 / r382752;
        double r382754 = r382749 - r382753;
        double r382755 = y;
        double r382756 = 3.0;
        double r382757 = sqrt(r382750);
        double r382758 = r382756 * r382757;
        double r382759 = r382755 / r382758;
        double r382760 = r382754 - r382759;
        return r382760;
}
↓
double f(double x, double y) {
        double r382761 = 1.0;
        double r382762 = cbrt(r382761);
        double r382763 = r382762 * r382762;
        double r382764 = 1.0;
        double r382765 = x;
        double r382766 = r382764 / r382765;
        double r382767 = 9.0;
        double r382768 = cbrt(r382767);
        double r382769 = r382766 / r382768;
        double r382770 = r382768 * r382768;
        double r382771 = r382761 / r382770;
        double r382772 = r382769 * r382771;
        double r382773 = -r382772;
        double r382774 = fma(r382763, r382762, r382773);
        double r382775 = -1.0;
        double r382776 = r382768 * r382765;
        double r382777 = r382775 / r382776;
        double r382778 = r382777 + r382769;
        double r382779 = r382771 * r382778;
        double r382780 = y;
        double r382781 = 3.0;
        double r382782 = r382780 / r382781;
        double r382783 = sqrt(r382765);
        double r382784 = r382782 / r382783;
        double r382785 = r382779 - r382784;
        double r382786 = r382774 + r382785;
        return r382786;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.2

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{\color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  6. Applied div-inv0.2

    \[\leadsto \left(1 - \frac{\color{blue}{1 \cdot \frac{1}{x}}}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  7. Applied times-frac0.3

    \[\leadsto \left(1 - \color{blue}{\frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  8. Applied add-cube-cbrt0.3

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  9. Applied prod-diff0.3

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \mathsf{fma}\left(-\frac{\frac{1}{x}}{\sqrt[3]{9}}, \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}, \frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right)\right)} - \frac{y}{3 \cdot \sqrt{x}}\]

  10. Applied associate--l+0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \left(\mathsf{fma}\left(-\frac{\frac{1}{x}}{\sqrt[3]{9}}, \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}, \frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\right)}\]

  11. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \color{blue}{\left(\frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \left(\frac{-1}{\sqrt[3]{9} \cdot x} + \frac{\frac{1}{x}}{\sqrt[3]{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)}\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\frac{\frac{1}{x}}{\sqrt[3]{9}} \cdot \frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}}\right) + \left(\frac{1}{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \left(\frac{-1}{\sqrt[3]{9} \cdot x} + \frac{\frac{1}{x}}{\sqrt[3]{9}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))