Average Error: 0.2 → 0.3
Time: 5.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \frac{\sqrt[3]{1}}{9}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{x} \cdot \frac{\sqrt[3]{1}}{9}\right) - \frac{\frac{y}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r367439 = 1.0;
        double r367440 = x;
        double r367441 = 9.0;
        double r367442 = r367440 * r367441;
        double r367443 = r367439 / r367442;
        double r367444 = r367439 - r367443;
        double r367445 = y;
        double r367446 = 3.0;
        double r367447 = sqrt(r367440);
        double r367448 = r367446 * r367447;
        double r367449 = r367445 / r367448;
        double r367450 = r367444 - r367449;
        return r367450;
}
↓
double f(double x, double y) {
        double r367451 = 1.0;
        double r367452 = cbrt(r367451);
        double r367453 = r367452 * r367452;
        double r367454 = x;
        double r367455 = r367453 / r367454;
        double r367456 = 9.0;
        double r367457 = r367452 / r367456;
        double r367458 = r367455 * r367457;
        double r367459 = r367451 - r367458;
        double r367460 = y;
        double r367461 = 3.0;
        double r367462 = r367460 / r367461;
        double r367463 = sqrt(r367454);
        double r367464 = r367462 / r367463;
        double r367465 = r367459 - r367464;
        return r367465;
}

Error

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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

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

  5. Applied add-cube-cbrt0.2

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

  6. Applied times-frac0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019322 
(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)))))