Average Error: 0.2 → 0.3
Time: 5.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt{x}}}{\sqrt[3]{3}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt{x}}}{\sqrt[3]{3}}
double f(double x, double y) {
        double r422473 = 1.0;
        double r422474 = x;
        double r422475 = 9.0;
        double r422476 = r422474 * r422475;
        double r422477 = r422473 / r422476;
        double r422478 = r422473 - r422477;
        double r422479 = y;
        double r422480 = 3.0;
        double r422481 = sqrt(r422474);
        double r422482 = r422480 * r422481;
        double r422483 = r422479 / r422482;
        double r422484 = r422478 - r422483;
        return r422484;
}
↓
double f(double x, double y) {
        double r422485 = 1.0;
        double r422486 = x;
        double r422487 = r422485 / r422486;
        double r422488 = 9.0;
        double r422489 = r422487 / r422488;
        double r422490 = r422485 - r422489;
        double r422491 = 1.0;
        double r422492 = cbrt(r422491);
        double r422493 = r422492 * r422492;
        double r422494 = 3.0;
        double r422495 = cbrt(r422494);
        double r422496 = r422495 * r422495;
        double r422497 = r422493 / r422496;
        double r422498 = y;
        double r422499 = sqrt(r422486);
        double r422500 = r422498 / r422499;
        double r422501 = r422500 / r422495;
        double r422502 = r422497 * r422501;
        double r422503 = r422490 - r422502;
        return r422503;
}

Error

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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-frac0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied add-cube-cbrt0.2

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

  10. Applied times-frac0.4

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

  11. Applied associate-*l*0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 
(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)))))