Average Error: 0.2 → 0.3
Time: 8.6s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \frac{\sqrt[3]{1}}{9}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \frac{\sqrt[3]{1}}{9}\right) - y \cdot \frac{1}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r382736 = 1.0;
        double r382737 = x;
        double r382738 = 9.0;
        double r382739 = r382737 * r382738;
        double r382740 = r382736 / r382739;
        double r382741 = r382736 - r382740;
        double r382742 = y;
        double r382743 = 3.0;
        double r382744 = sqrt(r382737);
        double r382745 = r382743 * r382744;
        double r382746 = r382742 / r382745;
        double r382747 = r382741 - r382746;
        return r382747;
}
↓
double f(double x, double y) {
        double r382748 = 1.0;
        double r382749 = cbrt(r382748);
        double r382750 = r382749 * r382749;
        double r382751 = x;
        double r382752 = r382750 / r382751;
        double r382753 = 9.0;
        double r382754 = r382749 / r382753;
        double r382755 = r382752 * r382754;
        double r382756 = r382748 - r382755;
        double r382757 = y;
        double r382758 = 1.0;
        double r382759 = 3.0;
        double r382760 = sqrt(r382751);
        double r382761 = r382759 * r382760;
        double r382762 = r382758 / r382761;
        double r382763 = r382757 * r382762;
        double r382764 = r382756 - r382763;
        return r382764;
}

Error

Bits error versus x

Bits error versus y

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Results

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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 add-cube-cbrt0.2

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

  4. Applied times-frac0.3

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

  6. Applied div-inv0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020083 +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)))))