Average Error: 0.4 → 0.5
Time: 20.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{\sqrt{\frac{1}{9}}}{\sqrt{x}} \cdot \frac{\sqrt{\frac{1}{9}}}{\sqrt{x}}\right) - 1\right)\]
\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{\sqrt{\frac{1}{9}}}{\sqrt{x}} \cdot \frac{\sqrt{\frac{1}{9}}}{\sqrt{x}}\right) - 1\right)
double f(double x, double y) {
        double r358794 = 3.0;
        double r358795 = x;
        double r358796 = sqrt(r358795);
        double r358797 = r358794 * r358796;
        double r358798 = y;
        double r358799 = 1.0;
        double r358800 = 9.0;
        double r358801 = r358795 * r358800;
        double r358802 = r358799 / r358801;
        double r358803 = r358798 + r358802;
        double r358804 = r358803 - r358799;
        double r358805 = r358797 * r358804;
        return r358805;
}
↓
double f(double x, double y) {
        double r358806 = 3.0;
        double r358807 = x;
        double r358808 = sqrt(r358807);
        double r358809 = r358806 * r358808;
        double r358810 = y;
        double r358811 = 1.0;
        double r358812 = 9.0;
        double r358813 = r358811 / r358812;
        double r358814 = sqrt(r358813);
        double r358815 = r358814 / r358808;
        double r358816 = r358815 * r358815;
        double r358817 = r358810 + r358816;
        double r358818 = r358817 - r358811;
        double r358819 = r358809 * r358818;
        return r358819;
}

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 *-un-lft-identity0.4

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

  4. Applied times-frac0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied times-frac0.5

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

  10. Applied unswap-sqr0.5

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

  11. Simplified0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020045 
(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)))