Average Error: 0.2 → 0.2
Time: 26.4s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{9}}}{\left|\sqrt[3]{9}\right|} \cdot \frac{\sqrt[3]{1}}{x}\right) \cdot \frac{1}{\sqrt{\sqrt[3]{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{9}}}{\left|\sqrt[3]{9}\right|} \cdot \frac{\sqrt[3]{1}}{x}\right) \cdot \frac{1}{\sqrt{\sqrt[3]{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r277045 = 1.0;
        double r277046 = x;
        double r277047 = 9.0;
        double r277048 = r277046 * r277047;
        double r277049 = r277045 / r277048;
        double r277050 = r277045 - r277049;
        double r277051 = y;
        double r277052 = 3.0;
        double r277053 = sqrt(r277046);
        double r277054 = r277052 * r277053;
        double r277055 = r277051 / r277054;
        double r277056 = r277050 - r277055;
        return r277056;
}
↓
double f(double x, double y) {
        double r277057 = 1.0;
        double r277058 = cbrt(r277057);
        double r277059 = r277058 * r277058;
        double r277060 = 9.0;
        double r277061 = sqrt(r277060);
        double r277062 = r277059 / r277061;
        double r277063 = cbrt(r277060);
        double r277064 = fabs(r277063);
        double r277065 = r277062 / r277064;
        double r277066 = x;
        double r277067 = r277058 / r277066;
        double r277068 = r277065 * r277067;
        double r277069 = 1.0;
        double r277070 = sqrt(r277063);
        double r277071 = r277069 / r277070;
        double r277072 = r277068 * r277071;
        double r277073 = r277057 - r277072;
        double r277074 = y;
        double r277075 = 3.0;
        double r277076 = sqrt(r277066);
        double r277077 = r277075 * r277076;
        double r277078 = r277074 / r277077;
        double r277079 = r277073 - r277078;
        return r277079;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-/r*0.2

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

  6. Simplified0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied sqrt-prod0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied times-frac0.3

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

  12. Applied times-frac0.2

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

  13. Simplified0.2

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

  15. Applied div-inv0.2

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

  16. Applied associate-*r*0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 +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)))))