Average Error: 0.2 → 0.2
Time: 11.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \frac{\frac{1}{3}}{\sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - y \cdot \frac{\frac{1}{3}}{\sqrt{x}}
double f(double x, double y) {
        double r455284 = 1.0;
        double r455285 = x;
        double r455286 = 9.0;
        double r455287 = r455285 * r455286;
        double r455288 = r455284 / r455287;
        double r455289 = r455284 - r455288;
        double r455290 = y;
        double r455291 = 3.0;
        double r455292 = sqrt(r455285);
        double r455293 = r455291 * r455292;
        double r455294 = r455290 / r455293;
        double r455295 = r455289 - r455294;
        return r455295;
}
↓
double f(double x, double y) {
        double r455296 = 1.0;
        double r455297 = x;
        double r455298 = 9.0;
        double r455299 = r455297 * r455298;
        double r455300 = r455296 / r455299;
        double r455301 = r455296 - r455300;
        double r455302 = y;
        double r455303 = 1.0;
        double r455304 = 3.0;
        double r455305 = r455303 / r455304;
        double r455306 = sqrt(r455297);
        double r455307 = r455305 / r455306;
        double r455308 = r455302 * r455307;
        double r455309 = r455301 - r455308;
        return r455309;
}

Error

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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 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 *-un-lft-identity0.2

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

  6. Applied sqrt-prod0.2

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

  7. Applied div-inv0.2

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

  8. Applied times-frac0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019350 +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)))))