Average Error: 29.5 → 0.2
Time: 4.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}
double f(double x) {
        double r196635 = x;
        double r196636 = 1.0;
        double r196637 = r196635 + r196636;
        double r196638 = sqrt(r196637);
        double r196639 = sqrt(r196635);
        double r196640 = r196638 - r196639;
        return r196640;
}
↓
double f(double x) {
        double r196641 = 1.0;
        double r196642 = 0.0;
        double r196643 = r196641 + r196642;
        double r196644 = x;
        double r196645 = r196644 + r196641;
        double r196646 = sqrt(r196645);
        double r196647 = sqrt(r196646);
        double r196648 = sqrt(r196644);
        double r196649 = fma(r196647, r196647, r196648);
        double r196650 = r196643 / r196649;
        return r196650;
}

Error

Bits error versus x

Target

Original29.5
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.3

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1 + 0}{\sqrt{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} + \sqrt{x}}\]

  7. Applied sqrt-prod0.3

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}} + \sqrt{x}}\]

  8. Applied fma-def0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}}\]

  9. Final simplification0.2

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x + 1}}, \sqrt{x}\right)}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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