Average Error: 19.8 → 0.5
Time: 4.5s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{\frac{1}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{\frac{1}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x} \cdot \sqrt{1 + x}}
double code(double x) {
	return ((double) (((double) (1.0 / ((double) sqrt(x)))) - ((double) (1.0 / ((double) sqrt(((double) (x + 1.0))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 * ((double) (((double) (1.0 / ((double) sqrt(((double) (((double) sqrt(x)) + ((double) sqrt(((double) (1.0 + x)))))))))) / ((double) sqrt(((double) (((double) sqrt(x)) + ((double) sqrt(((double) (1.0 + x)))))))))))) / ((double) (((double) sqrt(x)) * ((double) sqrt(((double) (1.0 + x))))))));
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  5. Simplified19.8

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

  7. Applied flip--19.6

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

  8. Simplified0.4

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied associate-/r*0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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