Average Error: 30.0 → 0.2
Time: 7.1s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{-0.5}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{-0.5}
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (pow (* (+ (sqrt (+ 1.0 x)) (sqrt x)) (+ (sqrt (+ 1.0 x)) (sqrt x))) -0.5))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	return pow(((sqrt(1.0 + x) + sqrt(x)) * (sqrt(1.0 + x) + sqrt(x))), -0.5);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--_binary64_1232929.8

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt_binary64_123760.3

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

  9. Applied inv-pow_binary64_124390.3

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

  10. Applied sqrt-pow1_binary64_123720.3

    \[\leadsto \sqrt{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}}\]

  11. Applied inv-pow_binary64_124390.3

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1}}} \cdot {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  12. Applied sqrt-pow1_binary64_123720.3

    \[\leadsto \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}\]

  13. Applied pow-prod-down_binary64_124250.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(\frac{-1}{2}\right)}}\]

  14. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{-0.5}\]

Reproduce

herbie shell --seed 2021044 
(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)))