Average Error: 29.4 → 0.2
Time: 4.9s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]
\sqrt{x + 1} - \sqrt{x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)
double f(double x) {
        double r150946 = x;
        double r150947 = 1.0;
        double r150948 = r150946 + r150947;
        double r150949 = sqrt(r150948);
        double r150950 = sqrt(r150946);
        double r150951 = r150949 - r150950;
        return r150951;
}
↓
double f(double x) {
        double r150952 = 1.0;
        double r150953 = 0.0;
        double r150954 = r150952 + r150953;
        double r150955 = x;
        double r150956 = r150955 + r150952;
        double r150957 = sqrt(r150956);
        double r150958 = sqrt(r150955);
        double r150959 = r150957 + r150958;
        double r150960 = r150954 / r150959;
        double r150961 = expm1(r150960);
        double r150962 = log1p(r150961);
        return r150962;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.4

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

  3. Applied flip--29.1

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied log1p-expm1-u0.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)}\]

  7. Final simplification0.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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