Average Error: 0.2 → 0.2
Time: 26.1s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{1}{\frac{\sqrt{x}}{\frac{y}{3}}}
double f(double x, double y) {
        double r291435 = 1.0;
        double r291436 = x;
        double r291437 = 9.0;
        double r291438 = r291436 * r291437;
        double r291439 = r291435 / r291438;
        double r291440 = r291435 - r291439;
        double r291441 = y;
        double r291442 = 3.0;
        double r291443 = sqrt(r291436);
        double r291444 = r291442 * r291443;
        double r291445 = r291441 / r291444;
        double r291446 = r291440 - r291445;
        return r291446;
}
↓
double f(double x, double y) {
        double r291447 = 1.0;
        double r291448 = x;
        double r291449 = r291447 / r291448;
        double r291450 = 9.0;
        double r291451 = r291449 / r291450;
        double r291452 = r291447 - r291451;
        double r291453 = 1.0;
        double r291454 = sqrt(r291448);
        double r291455 = y;
        double r291456 = 3.0;
        double r291457 = r291455 / r291456;
        double r291458 = r291454 / r291457;
        double r291459 = r291453 / r291458;
        double r291460 = r291452 - r291459;
        return r291460;
}

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 associate-/r*0.2

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

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

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

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

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

  9. Applied times-frac0.2

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

  10. Applied associate-/l*0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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)))))