Average Error: 30.0 → 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 r4675014 = x;
        double r4675015 = 1.0;
        double r4675016 = r4675014 + r4675015;
        double r4675017 = sqrt(r4675016);
        double r4675018 = sqrt(r4675014);
        double r4675019 = r4675017 - r4675018;
        return r4675019;
}
↓
double f(double x) {
        double r4675020 = 1.0;
        double r4675021 = x;
        double r4675022 = r4675021 + r4675020;
        double r4675023 = sqrt(r4675022);
        double r4675024 = sqrt(r4675021);
        double r4675025 = r4675023 + r4675024;
        double r4675026 = r4675020 / r4675025;
        double r4675027 = log1p(r4675026);
        double r4675028 = expm1(r4675027);
        return r4675028;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

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

  3. Applied flip--29.9

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

  4. Simplified29.4

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

  5. Simplified29.4

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
  6. Using strategy rm

  7. Applied expm1-log1p-u29.4

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

  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 2019168 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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