Average Error: 0.4 → 0.4
Time: 3.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r432919 = 3.0;
        double r432920 = x;
        double r432921 = sqrt(r432920);
        double r432922 = r432919 * r432921;
        double r432923 = y;
        double r432924 = 1.0;
        double r432925 = 9.0;
        double r432926 = r432920 * r432925;
        double r432927 = r432924 / r432926;
        double r432928 = r432923 + r432927;
        double r432929 = r432928 - r432924;
        double r432930 = r432922 * r432929;
        return r432930;
}
↓
double f(double x, double y) {
        double r432931 = 3.0;
        double r432932 = 0.1111111111111111;
        double r432933 = x;
        double r432934 = r432932 / r432933;
        double r432935 = y;
        double r432936 = 1.0;
        double r432937 = r432935 - r432936;
        double r432938 = r432934 + r432937;
        double r432939 = r432931 * r432938;
        double r432940 = sqrt(r432933);
        double r432941 = r432939 * r432940;
        return r432941;
}

Error

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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. Taylor expanded around 0 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right)} - 1\right)\]
  3. Using strategy rm

  4. Applied pow10.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{{\left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)}^{1}}\]

  5. Applied pow10.4

    \[\leadsto \left(3 \cdot \color{blue}{{\left(\sqrt{x}\right)}^{1}}\right) \cdot {\left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  6. Applied pow10.4

    \[\leadsto \left(\color{blue}{{3}^{1}} \cdot {\left(\sqrt{x}\right)}^{1}\right) \cdot {\left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  7. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(3 \cdot \sqrt{x}\right)}^{1}} \cdot {\left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)}^{1}\]

  8. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right) - 1\right)\right)}^{1}}\]

  9. Simplified0.4

    \[\leadsto {\color{blue}{\left(\left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)\right) \cdot \sqrt{x}\right)}}^{1}\]

  10. Final simplification0.4

    \[\leadsto \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)\right) \cdot \sqrt{x}\]

Reproduce

herbie shell --seed 2020025 
(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)))