Average Error: 30.2 → 0.2
Time: 30.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
double f(double x) {
        double r8188563 = x;
        double r8188564 = 1.0;
        double r8188565 = r8188563 + r8188564;
        double r8188566 = sqrt(r8188565);
        double r8188567 = sqrt(r8188563);
        double r8188568 = r8188566 - r8188567;
        return r8188568;
}
↓
double f(double x) {
        double r8188569 = 1.0;
        double r8188570 = x;
        double r8188571 = r8188570 + r8188569;
        double r8188572 = sqrt(r8188571);
        double r8188573 = sqrt(r8188570);
        double r8188574 = r8188572 + r8188573;
        double r8188575 = r8188569 / r8188574;
        return r8188575;
}


\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.0

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

  4. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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