Average Error: 0.2 → 0.3
Time: 10.6s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y \cdot \frac{1}{\sqrt{x}}}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y \cdot \frac{1}{\sqrt{x}}}{3}
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 * ((double) (1.0 / ((double) sqrt(x)))))) / 3.0))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.3
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

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 *-commutative0.2

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

  4. Applied associate-/r*0.3

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

  6. Applied div-inv0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020114 +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)))))