Average Error: 0.2 → 0.2
Time: 16.5s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\frac{3}{\frac{y}{\sqrt{x}}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\frac{3}{\frac{y}{\sqrt{x}}}}
double f(double x, double y) {
        double r238341 = 1.0;
        double r238342 = x;
        double r238343 = 9.0;
        double r238344 = r238342 * r238343;
        double r238345 = r238341 / r238344;
        double r238346 = r238341 - r238345;
        double r238347 = y;
        double r238348 = 3.0;
        double r238349 = sqrt(r238342);
        double r238350 = r238348 * r238349;
        double r238351 = r238347 / r238350;
        double r238352 = r238346 - r238351;
        return r238352;
}
↓
double f(double x, double y) {
        double r238353 = 1.0;
        double r238354 = x;
        double r238355 = 9.0;
        double r238356 = r238354 * r238355;
        double r238357 = r238353 / r238356;
        double r238358 = r238353 - r238357;
        double r238359 = 1.0;
        double r238360 = 3.0;
        double r238361 = y;
        double r238362 = sqrt(r238354);
        double r238363 = r238361 / r238362;
        double r238364 = r238360 / r238363;
        double r238365 = r238359 / r238364;
        double r238366 = r238358 - r238365;
        return r238366;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

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 clear-num0.2

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

  5. Applied add-sqr-sqrt0.2

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

  6. Applied associate-/l*0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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)))))