Average Error: 0.2 → 0.1
Time: 9.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}
double code(double x) {
	return ((double) (((double) (6.0 * ((double) (x - 1.0)))) / ((double) (((double) (x + 1.0)) + ((double) (4.0 * ((double) sqrt(x))))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) (((double) sqrt(x)) + ((double) sqrt(1.0)))) / ((double) sqrt(((double) fma(((double) sqrt(x)), 4.0, ((double) (x + 1.0)))))))) * ((double) (((double) (((double) sqrt(x)) - ((double) sqrt(1.0)))) / ((double) (((double) sqrt(((double) fma(((double) sqrt(x)), 4.0, ((double) (x + 1.0)))))) / 6.0))))));
}

Error

Bits error versus x

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Results

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Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.1

    \[\leadsto \frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{\color{blue}{1 \cdot 6}}}\]

  5. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{x - 1}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}}{1 \cdot 6}}\]

  6. Applied times-frac0.3

    \[\leadsto \frac{x - 1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}}\]

  7. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{x - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  8. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{1} \cdot \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  9. Applied difference-of-squares0.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt{x} + \sqrt{1}\right) \cdot \left(\sqrt{x} - \sqrt{1}\right)}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  10. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{\sqrt{x} + \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  12. Final simplification0.1

    \[\leadsto \frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))