Average Error: 29.5 → 0.2
Time: 18.3s
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 r4030786 = x;
        double r4030787 = 1.0;
        double r4030788 = r4030786 + r4030787;
        double r4030789 = sqrt(r4030788);
        double r4030790 = sqrt(r4030786);
        double r4030791 = r4030789 - r4030790;
        return r4030791;
}
↓
double f(double x) {
        double r4030792 = 1.0;
        double r4030793 = x;
        double r4030794 = r4030793 + r4030792;
        double r4030795 = sqrt(r4030794);
        double r4030796 = sqrt(r4030793);
        double r4030797 = r4030795 + r4030796;
        double r4030798 = r4030792 / r4030797;
        double r4030799 = log1p(r4030798);
        double r4030800 = expm1(r4030799);
        return r4030800;
}

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)))