Average Error: 19.5 → 0.1
Time: 10.2s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[{x}^{-0.5} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x + 1\right)} \]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
{x}^{-0.5} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x + 1\right)}

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

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 19.5

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

  2. Applied frac-sub_binary6419.5

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

  3. Simplified19.5

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

  4. Simplified19.5

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

  5. Applied flip--_binary6419.3

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

  6. Simplified0.4

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

  7. Applied *-un-lft-identity_binary640.4

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

  8. Applied add-cube-cbrt_binary640.4

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

  9. Applied times-frac_binary640.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{1 + x}} \]

  10. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\sqrt{x}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}} \]

  11. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{x}}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}} \]

  12. Simplified0.3

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

  13. Applied pow1_binary640.3

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

  14. Applied sqrt-pow1_binary640.3

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

  15. Applied pow-flip_binary640.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(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)))))