Average Error: 30.6 → 0.3
Time: 4.0s
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 r685456 = x;
        double r685457 = 1.0;
        double r685458 = r685456 + r685457;
        double r685459 = sqrt(r685458);
        double r685460 = sqrt(r685456);
        double r685461 = r685459 - r685460;
        return r685461;
}
↓
double f(double x) {
        double r685462 = 1.0;
        double r685463 = 0.0;
        double r685464 = r685462 + r685463;
        double r685465 = x;
        double r685466 = r685465 + r685462;
        double r685467 = cbrt(r685466);
        double r685468 = r685467 * r685467;
        double r685469 = sqrt(r685468);
        double r685470 = sqrt(r685467);
        double r685471 = sqrt(r685465);
        double r685472 = fma(r685469, r685470, r685471);
        double r685473 = r685464 / r685472;
        return r685473;
}

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 "Main:bigenough3 from C"
  :precision binary64

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

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