Average Error: 0.2 → 0.2
Time: 3.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - {\left(\frac{\frac{y}{\sqrt{x}}}{3}\right)}^{1}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - {\left(\frac{\frac{y}{\sqrt{x}}}{3}\right)}^{1}
double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x))));
}
↓
double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - pow(((y / sqrt(x)) / 3.0), 1.0));
}

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 *-un-lft-identity0.2

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

  4. Applied times-frac0.3

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

  6. Applied pow10.3

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

  7. Applied pow10.3

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

  8. Applied pow-prod-down0.3

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020079 
(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)))))