Average Error: 0.4 → 0.4
Time: 15.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(-1\right) \cdot \sqrt{x}\right) \cdot 3 + \left(\left(\frac{\frac{1}{x}}{9} + y\right) \cdot \sqrt{x}\right) \cdot 3\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(-1\right) \cdot \sqrt{x}\right) \cdot 3 + \left(\left(\frac{\frac{1}{x}}{9} + y\right) \cdot \sqrt{x}\right) \cdot 3
double f(double x, double y) {
        double r251835 = 3.0;
        double r251836 = x;
        double r251837 = sqrt(r251836);
        double r251838 = r251835 * r251837;
        double r251839 = y;
        double r251840 = 1.0;
        double r251841 = 9.0;
        double r251842 = r251836 * r251841;
        double r251843 = r251840 / r251842;
        double r251844 = r251839 + r251843;
        double r251845 = r251844 - r251840;
        double r251846 = r251838 * r251845;
        return r251846;
}
↓
double f(double x, double y) {
        double r251847 = 1.0;
        double r251848 = -r251847;
        double r251849 = x;
        double r251850 = sqrt(r251849);
        double r251851 = r251848 * r251850;
        double r251852 = 3.0;
        double r251853 = r251851 * r251852;
        double r251854 = r251847 / r251849;
        double r251855 = 9.0;
        double r251856 = r251854 / r251855;
        double r251857 = y;
        double r251858 = r251856 + r251857;
        double r251859 = r251858 * r251850;
        double r251860 = r251859 * r251852;
        double r251861 = r251853 + r251860;
        return r251861;
}

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 sub-neg0.4

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

  4. Applied distribute-lft-in0.4

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied *-commutative0.4

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

  10. Applied associate-*r*0.4

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

  11. Simplified0.5

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))