Average Error: 0.4 → 0.5
Time: 5.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r569489 = 3.0;
        double r569490 = x;
        double r569491 = sqrt(r569490);
        double r569492 = r569489 * r569491;
        double r569493 = y;
        double r569494 = 1.0;
        double r569495 = 9.0;
        double r569496 = r569490 * r569495;
        double r569497 = r569494 / r569496;
        double r569498 = r569493 + r569497;
        double r569499 = r569498 - r569494;
        double r569500 = r569492 * r569499;
        return r569500;
}
↓
double f(double x, double y) {
        double r569501 = 3.0;
        double r569502 = y;
        double r569503 = r569501 * r569502;
        double r569504 = 0.3333333333333333;
        double r569505 = 1.0;
        double r569506 = x;
        double r569507 = r569505 / r569506;
        double r569508 = r569504 * r569507;
        double r569509 = r569503 + r569508;
        double r569510 = r569509 - r569501;
        double r569511 = sqrt(r569506);
        double r569512 = r569510 * r569511;
        return r569512;
}

Error

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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 pow10.4

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

  7. Applied pow10.4

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

  8. Applied pow10.4

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

  9. Applied pow-prod-down0.4

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

  10. Applied pow-prod-down0.4

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

  11. Simplified0.5

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

  12. Taylor expanded around 0 0.5

    \[\leadsto {\left(\color{blue}{\left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right)} \cdot \sqrt{x}\right)}^{1}\]

  13. Final simplification0.5

    \[\leadsto \left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right) \cdot \sqrt{x}\]

Reproduce

herbie shell --seed 2019362 
(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)))