Average Error: 0.2 → 0.3
Time: 16.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\sqrt{1}}{x} \cdot \frac{\sqrt{1}}{9}\right) - \frac{1}{\frac{3 \cdot \sqrt{x}}{y}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\sqrt{1}}{x} \cdot \frac{\sqrt{1}}{9}\right) - \frac{1}{\frac{3 \cdot \sqrt{x}}{y}}
double f(double x, double y) {
        double r288490 = 1.0;
        double r288491 = x;
        double r288492 = 9.0;
        double r288493 = r288491 * r288492;
        double r288494 = r288490 / r288493;
        double r288495 = r288490 - r288494;
        double r288496 = y;
        double r288497 = 3.0;
        double r288498 = sqrt(r288491);
        double r288499 = r288497 * r288498;
        double r288500 = r288496 / r288499;
        double r288501 = r288495 - r288500;
        return r288501;
}
↓
double f(double x, double y) {
        double r288502 = 1.0;
        double r288503 = sqrt(r288502);
        double r288504 = x;
        double r288505 = r288503 / r288504;
        double r288506 = 9.0;
        double r288507 = r288503 / r288506;
        double r288508 = r288505 * r288507;
        double r288509 = r288502 - r288508;
        double r288510 = 1.0;
        double r288511 = 3.0;
        double r288512 = sqrt(r288504);
        double r288513 = r288511 * r288512;
        double r288514 = y;
        double r288515 = r288513 / r288514;
        double r288516 = r288510 / r288515;
        double r288517 = r288509 - r288516;
        return r288517;
}

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 add-sqr-sqrt0.2

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

  4. Applied times-frac0.3

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

  6. Applied clear-num0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019208 
(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)))))