Average Error: 29.3 → 0.2
Time: 14.1s
Precision: 64
\[\sqrt{x + 1.0} - \sqrt{x}\]
↓
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]
\sqrt{x + 1.0} - \sqrt{x}
↓
\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}
double f(double x) {
        double r23985973 = x;
        double r23985974 = 1.0;
        double r23985975 = r23985973 + r23985974;
        double r23985976 = sqrt(r23985975);
        double r23985977 = sqrt(r23985973);
        double r23985978 = r23985976 - r23985977;
        return r23985978;
}
↓
double f(double x) {
        double r23985979 = 1.0;
        double r23985980 = x;
        double r23985981 = r23985980 + r23985979;
        double r23985982 = sqrt(r23985981);
        double r23985983 = sqrt(r23985980);
        double r23985984 = r23985982 + r23985983;
        double r23985985 = r23985979 / r23985984;
        return r23985985;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

    \[\sqrt{x + 1.0} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.1

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1.0} \cdot \sqrt{x + 1.0} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1.0} + \sqrt{x}}}\]

  4. Simplified28.8

    \[\leadsto \frac{\color{blue}{x + \left(1.0 - x\right)}}{\sqrt{x + 1.0} + \sqrt{x}}\]

  5. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{1.0}}{\sqrt{x + 1.0} + \sqrt{x}}\]

  6. Final simplification0.2

    \[\leadsto \frac{1.0}{\sqrt{x + 1.0} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "Main:bigenough3 from C"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))