Average Error: 0.2 → 0.2
Time: 2.4min
Precision: binary64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[1 - \left(\frac{\frac{\sqrt[3]{1}}{\frac{x}{\sqrt[3]{1}}}}{9} + \frac{\frac{y}{3}}{\sqrt{x}}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
1 - \left(\frac{\frac{\sqrt[3]{1}}{\frac{x}{\sqrt[3]{1}}}}{9} + \frac{\frac{y}{3}}{\sqrt{x}}\right)
(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))
↓
(FPCore (x y)
 :precision binary64
 (- 1.0 (+ (/ (/ (cbrt 1.0) (/ x (cbrt 1.0))) 9.0) (/ (/ y 3.0) (sqrt x)))))
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 - (((cbrt(1.0) / (x / cbrt(1.0))) / 9.0) + ((y / 3.0) / sqrt(x)));
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

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. Simplified0.2

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

  4. Applied associate-/r*_binary64_68420.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2021176 
(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)))))