Average Error: 29.5 → 0.2
Time: 33.7s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}}, \sqrt{1 + x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}}, \sqrt{1 + x}\right)}
double f(double x) {
        double r5126618 = x;
        double r5126619 = 1.0;
        double r5126620 = r5126618 + r5126619;
        double r5126621 = sqrt(r5126620);
        double r5126622 = sqrt(r5126618);
        double r5126623 = r5126621 - r5126622;
        return r5126623;
}
↓
double f(double x) {
        double r5126624 = 1.0;
        double r5126625 = x;
        double r5126626 = sqrt(r5126625);
        double r5126627 = sqrt(r5126626);
        double r5126628 = r5126624 + r5126625;
        double r5126629 = sqrt(r5126628);
        double r5126630 = fma(r5126627, r5126627, r5126629);
        double r5126631 = r5126624 / r5126630;
        return r5126631;
}

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--29.4

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

  4. Simplified28.9

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

  5. Simplified28.9

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity28.9

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

  8. Applied associate-/r*28.9

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

  9. Simplified0.2

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

  11. Applied add-sqr-sqrt0.2

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

  12. Applied sqrt-prod0.2

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

  13. Applied fma-def0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))