Average Error: 0.2 → 0.2
Time: 4.2s
Precision: binary64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\sqrt{x}}}{3}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\sqrt{x}}}{3}
(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))
↓
(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ (/ y (sqrt x)) 3.0)))
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))) - ((y / 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
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

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 clear-num_binary64_41200.2

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

  4. Simplified0.2

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

  6. Applied *-un-lft-identity_binary64_41210.2

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

  7. Applied sqrt-prod_binary64_41360.2

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

  8. Applied *-un-lft-identity_binary64_41210.2

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

  9. Applied times-frac_binary64_41270.2

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

  10. Applied *-un-lft-identity_binary64_41210.2

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

  11. Applied times-frac_binary64_41270.2

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

  12. Applied add-sqr-sqrt_binary64_41420.2

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

  13. Applied times-frac_binary64_41270.2

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

  14. Simplified0.2

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020281 
(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)))))