Average Error: 30.6 → 0.3
Time: 4.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}
double f(double x) {
        double r97717 = x;
        double r97718 = 1.0;
        double r97719 = r97717 + r97718;
        double r97720 = sqrt(r97719);
        double r97721 = sqrt(r97717);
        double r97722 = r97720 - r97721;
        return r97722;
}
↓
double f(double x) {
        double r97723 = 1.0;
        double r97724 = 0.0;
        double r97725 = r97723 + r97724;
        double r97726 = x;
        double r97727 = r97726 + r97723;
        double r97728 = cbrt(r97727);
        double r97729 = r97728 * r97728;
        double r97730 = sqrt(r97729);
        double r97731 = sqrt(r97728);
        double r97732 = sqrt(r97726);
        double r97733 = fma(r97730, r97731, r97732);
        double r97734 = r97725 / r97733;
        return r97734;
}

Error

Bits error versus x

Target

Original30.6
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  4. Simplified0.2

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

  6. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1 + 0}{\sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} + \sqrt{x}}\]

  7. Applied sqrt-prod0.3

    \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} + \sqrt{x}}\]

  8. Applied fma-def0.3

    \[\leadsto \frac{1 + 0}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}}\]

  9. Final simplification0.3

    \[\leadsto \frac{1 + 0}{\mathsf{fma}\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt{\sqrt[3]{x + 1}}, \sqrt{x}\right)}\]

Reproduce

herbie shell --seed 2020025 +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)))