Average Error: 0.4 → 0.4
Time: 4.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{\sqrt{x}}}{\sqrt{x} \cdot 9}\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\frac{1}{\sqrt{x}}}{\sqrt{x} \cdot 9}\right) - 1\right)
double f(double x, double y) {
        double r450058 = 3.0;
        double r450059 = x;
        double r450060 = sqrt(r450059);
        double r450061 = r450058 * r450060;
        double r450062 = y;
        double r450063 = 1.0;
        double r450064 = 9.0;
        double r450065 = r450059 * r450064;
        double r450066 = r450063 / r450065;
        double r450067 = r450062 + r450066;
        double r450068 = r450067 - r450063;
        double r450069 = r450061 * r450068;
        return r450069;
}
↓
double f(double x, double y) {
        double r450070 = 3.0;
        double r450071 = x;
        double r450072 = sqrt(r450071);
        double r450073 = r450070 * r450072;
        double r450074 = y;
        double r450075 = 1.0;
        double r450076 = r450075 / r450072;
        double r450077 = 9.0;
        double r450078 = r450072 * r450077;
        double r450079 = r450076 / r450078;
        double r450080 = r450074 + r450079;
        double r450081 = r450080 - r450075;
        double r450082 = r450073 * r450081;
        return r450082;
}

Error

Bits error versus x

Bits error versus y

Try it out

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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 *-un-lft-identity0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  12. Applied frac-times0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +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)))