Average Error: 30.0 → 0.2
Time: 24.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
double f(double x) {
        double r5422757 = x;
        double r5422758 = 1.0;
        double r5422759 = r5422757 + r5422758;
        double r5422760 = sqrt(r5422759);
        double r5422761 = sqrt(r5422757);
        double r5422762 = r5422760 - r5422761;
        return r5422762;
}
↓
double f(double x) {
        double r5422763 = 1.0;
        double r5422764 = x;
        double r5422765 = r5422763 + r5422764;
        double r5422766 = sqrt(r5422765);
        double r5422767 = sqrt(r5422764);
        double r5422768 = r5422766 + r5422767;
        double r5422769 = r5422768 * r5422768;
        double r5422770 = -0.5;
        double r5422771 = pow(r5422769, r5422770);
        return r5422771;
}


\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}

Error

Bits error versus x

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--29.8

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

  4. Taylor expanded around 0 0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied pow10.3

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

  9. Applied sqrt-pow10.3

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

  10. Applied pow10.3

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

  11. Applied sqrt-pow10.3

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

  12. Applied pow-prod-down0.2

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

  13. Applied pow-flip0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019102 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))