Average Error: 30.0 → 0.2
Time: 20.9s
Precision: binary64
Cost: 576
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{1 + x} + \sqrt{x}}
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	return 1.0 / (sqrt(1.0 + x) + sqrt(x));
}

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
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
Alternative 1
Accuracy0.3
Cost1216
\[\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 2
Accuracy64.0
Cost1728
\[\frac{1}{2 \cdot \left(\mathsf{NaN} - \frac{{\mathsf{NaN}}^{3}}{x \cdot x}\right) - \left(\frac{\mathsf{NaN}}{x} - 3 \cdot \frac{{\mathsf{NaN}}^{5}}{x \cdot x}\right)}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--_binary64_175829.8

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

  4. Simplified0.2

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

  6. Applied *-un-lft-identity_binary64_17830.2

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

  7. Applied sqrt-prod_binary64_17990.2

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

  8. Applied *-un-lft-identity_binary64_17830.2

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

  9. Applied sqrt-prod_binary64_17990.2

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

  10. Applied distribute-lft-out_binary64_17340.2

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

  11. Simplified0.2

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

  13. Applied *-un-lft-identity_binary64_17830.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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