Average Error: 0.4 → 0.4
Time: 45.0s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\left(\left(\frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{9}} - 1\right) + y\right) \cdot \sqrt{x}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\left(\left(\frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{9}} - 1\right) + y\right) \cdot \sqrt{x}\right)
double f(double x, double y) {
        double r289949 = 3.0;
        double r289950 = x;
        double r289951 = sqrt(r289950);
        double r289952 = r289949 * r289951;
        double r289953 = y;
        double r289954 = 1.0;
        double r289955 = 9.0;
        double r289956 = r289950 * r289955;
        double r289957 = r289954 / r289956;
        double r289958 = r289953 + r289957;
        double r289959 = r289958 - r289954;
        double r289960 = r289952 * r289959;
        return r289960;
}
↓
double f(double x, double y) {
        double r289961 = 3.0;
        double r289962 = 1.0;
        double r289963 = x;
        double r289964 = r289962 / r289963;
        double r289965 = 9.0;
        double r289966 = cbrt(r289965);
        double r289967 = r289966 * r289966;
        double r289968 = r289964 / r289967;
        double r289969 = r289968 / r289966;
        double r289970 = r289969 - r289962;
        double r289971 = y;
        double r289972 = r289970 + r289971;
        double r289973 = sqrt(r289963);
        double r289974 = r289972 * r289973;
        double r289975 = r289961 * r289974;
        return r289975;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

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

  3. Applied associate-/r*0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  8. Applied add-cube-cbrt0.4

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

  9. Applied associate-/r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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