Average Error: 29.5 → 0.2
Time: 4.2s
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 r606596 = x;
        double r606597 = 1.0;
        double r606598 = r606596 + r606597;
        double r606599 = sqrt(r606598);
        double r606600 = sqrt(r606596);
        double r606601 = r606599 - r606600;
        return r606601;
}
↓
double f(double x) {
        double r606602 = 1.0;
        double r606603 = 0.0;
        double r606604 = r606602 + r606603;
        double r606605 = x;
        double r606606 = r606605 + r606602;
        double r606607 = sqrt(r606606);
        double r606608 = sqrt(r606607);
        double r606609 = sqrt(r606605);
        double r606610 = fma(r606608, r606608, r606609);
        double r606611 = r606604 / r606610;
        return r606611;
}

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 "Main:bigenough3 from C"
  :precision binary64

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

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