Average Error: 19.8 → 0.3
Time: 15.0s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, \sqrt{x}\right)\\ \frac{\frac{\frac{1}{t_0}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}}{t_0} \end{array} \]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
t_0 := \mathsf{hypot}\left(x, \sqrt{x}\right)\\
\frac{\frac{\frac{1}{t_0}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}}{t_0}
\end{array}

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.8
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.8

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

  2. Applied flip--_binary6419.8

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

  3. Simplified19.8

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

  4. Simplified19.8

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

  5. Applied frac-sub_binary6419.1

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

  6. Simplified5.6

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

  7. Simplified5.6

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

  8. Applied *-un-lft-identity_binary645.6

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

  9. Applied add-sqr-sqrt_binary645.6

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

  10. Applied add-cube-cbrt_binary645.6

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, x\right)}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \]

  11. Applied times-frac_binary645.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \]

  12. Applied times-frac_binary645.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}} \]

  13. Simplified5.5

    \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}} \]

  14. Simplified0.4

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

  15. Applied associate-*l/_binary640.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]

Reproduce

herbie shell --seed 2021224 
(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)))))