Average Error: 0.2 → 0.2
Time: 16.7s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(0 \cdot \frac{\frac{1}{x}}{9} + \left(1 - \frac{\frac{1}{x}}{9}\right)\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(0 \cdot \frac{\frac{1}{x}}{9} + \left(1 - \frac{\frac{1}{x}}{9}\right)\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}
double f(double x, double y) {
        double r280070 = 1.0;
        double r280071 = x;
        double r280072 = 9.0;
        double r280073 = r280071 * r280072;
        double r280074 = r280070 / r280073;
        double r280075 = r280070 - r280074;
        double r280076 = y;
        double r280077 = 3.0;
        double r280078 = sqrt(r280071);
        double r280079 = r280077 * r280078;
        double r280080 = r280076 / r280079;
        double r280081 = r280075 - r280080;
        return r280081;
}
↓
double f(double x, double y) {
        double r280082 = 0.0;
        double r280083 = 1.0;
        double r280084 = x;
        double r280085 = r280083 / r280084;
        double r280086 = 9.0;
        double r280087 = r280085 / r280086;
        double r280088 = r280082 * r280087;
        double r280089 = r280083 - r280087;
        double r280090 = r280088 + r280089;
        double r280091 = 1.0;
        double r280092 = sqrt(r280084);
        double r280093 = y;
        double r280094 = 3.0;
        double r280095 = r280093 / r280094;
        double r280096 = r280092 / r280095;
        double r280097 = r280091 / r280096;
        double r280098 = r280090 - r280097;
        return r280098;
}

Error

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Bits error versus y

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Results

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Target

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

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

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied prod-diff0.5

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

  8. Simplified0.2

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

  9. Simplified0.2

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

  11. Applied clear-num0.2

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

  12. Final simplification0.2

    \[\leadsto \left(0 \cdot \frac{\frac{1}{x}}{9} + \left(1 - \frac{\frac{1}{x}}{9}\right)\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}\]

Reproduce

herbie shell --seed 2019323 +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)))))