Average Error: 29.5 → 0.2
Time: 18.6s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)
double f(double x) {
        double r3153266 = x;
        double r3153267 = 1.0;
        double r3153268 = r3153266 + r3153267;
        double r3153269 = sqrt(r3153268);
        double r3153270 = sqrt(r3153266);
        double r3153271 = r3153269 - r3153270;
        return r3153271;
}
↓
double f(double x) {
        double r3153272 = 1.0;
        double r3153273 = x;
        double r3153274 = r3153273 + r3153272;
        double r3153275 = sqrt(r3153274);
        double r3153276 = sqrt(r3153273);
        double r3153277 = r3153275 + r3153276;
        double r3153278 = r3153272 / r3153277;
        double r3153279 = log1p(r3153278);
        double r3153280 = expm1(r3153279);
        return r3153280;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. 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 expm1-log1p-u28.9

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 + x\right) - x}{\sqrt{x} + \sqrt{1 + x}}\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)}\right)\]

  9. Final simplification0.2

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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