Average Error: 0.4 → 0.5
Time: 4.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot y\right) \cdot \sqrt{x} + 3 \cdot \left(\left(\frac{\sqrt{1}}{x} \cdot \frac{\sqrt{1}}{9} - 1\right) \cdot \sqrt{x}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot y\right) \cdot \sqrt{x} + 3 \cdot \left(\left(\frac{\sqrt{1}}{x} \cdot \frac{\sqrt{1}}{9} - 1\right) \cdot \sqrt{x}\right)
double f(double x, double y) {
        double r343926 = 3.0;
        double r343927 = x;
        double r343928 = sqrt(r343927);
        double r343929 = r343926 * r343928;
        double r343930 = y;
        double r343931 = 1.0;
        double r343932 = 9.0;
        double r343933 = r343927 * r343932;
        double r343934 = r343931 / r343933;
        double r343935 = r343930 + r343934;
        double r343936 = r343935 - r343931;
        double r343937 = r343929 * r343936;
        return r343937;
}
↓
double f(double x, double y) {
        double r343938 = 3.0;
        double r343939 = y;
        double r343940 = r343938 * r343939;
        double r343941 = x;
        double r343942 = sqrt(r343941);
        double r343943 = r343940 * r343942;
        double r343944 = 1.0;
        double r343945 = sqrt(r343944);
        double r343946 = r343945 / r343941;
        double r343947 = 9.0;
        double r343948 = r343945 / r343947;
        double r343949 = r343946 * r343948;
        double r343950 = r343949 - r343944;
        double r343951 = r343950 * r343942;
        double r343952 = r343938 * r343951;
        double r343953 = r343943 + r343952;
        return r343953;
}

Error

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Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.5
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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-*l*0.4

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

  5. Applied associate--l+0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  11. Applied associate-*r*0.4

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

  13. Applied add-sqr-sqrt0.4

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

  14. Applied times-frac0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019354 +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)))