Average Error: 29.7 → 0.3
Time: 4.8s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}
double code(double x) {
	return ((double) (((double) sqrt(((double) (x + 1.0)))) - ((double) sqrt(x))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) (((double) (1.0 + 0.0)) / ((double) (((double) sqrt(((double) (x + 1.0)))) + ((double) sqrt(x)))))))) * ((double) sqrt(((double) (((double) (1.0 + 0.0)) / ((double) (((double) sqrt(((double) (x + 1.0)))) + ((double) sqrt(x))))))))));
}

Error

Bits error versus x

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Results

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Target

Original29.7
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.5

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\]

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}}\]

  7. Final simplification0.3

    \[\leadsto \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))