Average Error: 30.1 → 0.2
Time: 3.8s
Precision: binary64
\[\sqrt{x + 1} - \sqrt{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt{1 + x} + \sqrt{x}\\ {\left(t_0 \cdot t_0\right)}^{-0.5} \end{array} \]
\sqrt{x + 1} - \sqrt{x}
↓
\begin{array}{l}
t_0 := \sqrt{1 + x} + \sqrt{x}\\
{\left(t_0 \cdot t_0\right)}^{-0.5}
\end{array}

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt (+ 1.0 x)) (sqrt x)))) (pow (* t_0 t_0) -0.5)))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	double t_0 = sqrt(1.0 + x) + sqrt(x);
	return pow((t_0 * t_0), -0.5);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.1

    \[\sqrt{x + 1} - \sqrt{x} \]

  2. Applied flip--_binary6429.9

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

  3. Simplified0.2

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

  4. Simplified0.2

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

  5. Applied add-sqr-sqrt_binary640.3

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

  6. Applied inv-pow_binary640.3

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

  7. Applied sqrt-pow1_binary640.3

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

  8. Applied inv-pow_binary640.3

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

  9. Applied sqrt-pow1_binary640.3

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

  10. Applied pow-prod-down_binary640.2

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

  11. 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 1991292424 
(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)))