Average Error: 0.2 → 0.3
Time: 20.9s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - y \cdot \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt{x} \cdot \sqrt[3]{3}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - y \cdot \frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt{x} \cdot \sqrt[3]{3}}
double f(double x, double y) {
        double r28066329 = 1.0;
        double r28066330 = x;
        double r28066331 = 9.0;
        double r28066332 = r28066330 * r28066331;
        double r28066333 = r28066329 / r28066332;
        double r28066334 = r28066329 - r28066333;
        double r28066335 = y;
        double r28066336 = 3.0;
        double r28066337 = sqrt(r28066330);
        double r28066338 = r28066336 * r28066337;
        double r28066339 = r28066335 / r28066338;
        double r28066340 = r28066334 - r28066339;
        return r28066340;
}
↓
double f(double x, double y) {
        double r28066341 = 1.0;
        double r28066342 = x;
        double r28066343 = r28066341 / r28066342;
        double r28066344 = 9.0;
        double r28066345 = r28066343 / r28066344;
        double r28066346 = r28066341 - r28066345;
        double r28066347 = y;
        double r28066348 = 1.0;
        double r28066349 = 3.0;
        double r28066350 = cbrt(r28066349);
        double r28066351 = r28066350 * r28066350;
        double r28066352 = r28066348 / r28066351;
        double r28066353 = sqrt(r28066342);
        double r28066354 = r28066353 * r28066350;
        double r28066355 = r28066352 / r28066354;
        double r28066356 = r28066347 * r28066355;
        double r28066357 = r28066346 - r28066356;
        return r28066357;
}

Error

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Bits error versus y

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Results

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Target

Original0.2
Target0.3
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*0.3

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

  5. Applied associate-/r*0.2

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

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

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

  8. Applied sqrt-prod0.2

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

  9. Applied div-inv0.3

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  13. Applied add-cube-cbrt0.3

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

  14. Applied *-un-lft-identity0.3

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

  15. Applied times-frac0.4

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

  16. Applied associate-/l*0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

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

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

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