Average Error: 0.2 → 0.3
Time: 4.4s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}
double f(double x, double y) {
        double r404425 = 1.0;
        double r404426 = x;
        double r404427 = 9.0;
        double r404428 = r404426 * r404427;
        double r404429 = r404425 / r404428;
        double r404430 = r404425 - r404429;
        double r404431 = y;
        double r404432 = 3.0;
        double r404433 = sqrt(r404426);
        double r404434 = r404432 * r404433;
        double r404435 = r404431 / r404434;
        double r404436 = r404430 - r404435;
        return r404436;
}
↓
double f(double x, double y) {
        double r404437 = 1.0;
        double r404438 = x;
        double r404439 = 9.0;
        double r404440 = r404438 * r404439;
        double r404441 = r404437 / r404440;
        double r404442 = r404437 - r404441;
        double r404443 = 1.0;
        double r404444 = 3.0;
        double r404445 = r404443 / r404444;
        double r404446 = y;
        double r404447 = 0.5;
        double r404448 = pow(r404438, r404447);
        double r404449 = r404446 / r404448;
        double r404450 = r404445 * r404449;
        double r404451 = r404442 - r404450;
        return r404451;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

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

  3. Applied associate-/r*0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{3} \cdot \color{blue}{\frac{y}{{x}^{\frac{1}{2}}}}\]
  8. Using strategy rm

  9. Applied div-inv0.3

    \[\leadsto \left(1 - \frac{\color{blue}{1 \cdot \frac{1}{x}}}{9}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}\]

  10. Applied associate-/l*0.3

    \[\leadsto \left(1 - \color{blue}{\frac{1}{\frac{9}{\frac{1}{x}}}}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}\]

  11. Simplified0.3

    \[\leadsto \left(1 - \frac{1}{\color{blue}{x \cdot 9}}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}\]

  12. Final simplification0.3

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{3} \cdot \frac{y}{{x}^{\frac{1}{2}}}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))