Average Error: 0.2 → 0.2
Time: 3.2s
Precision: binary64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}
double code(double x, double y) {
	return ((double) (((double) (1.0 - ((double) (1.0 / ((double) (x * 9.0)))))) - ((double) (y / ((double) (3.0 * ((double) sqrt(x))))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (1.0 - ((double) (1.0 / ((double) (x * 9.0)))))) - ((double) (((double) (y / 3.0)) * ((double) pow(x, -0.5))))));
}

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 div-inv0.2

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

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

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

  6. Applied times-frac0.3

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

  7. Applied associate-*r*0.3

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

  8. Simplified0.2

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

  10. Applied pow1/20.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}}\]

  11. Applied pow-flip0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot \color{blue}{{x}^{\left(-\frac{1}{2}\right)}}\]

  12. Simplified0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\color{blue}{\frac{-1}{2}}}\]

  13. Final simplification0.2

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3} \cdot {x}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2020177 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))