Average Error: 29.3 → 0.2
Time: 17.5s
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 r16049915 = x;
        double r16049916 = 1.0;
        double r16049917 = r16049915 + r16049916;
        double r16049918 = sqrt(r16049917);
        double r16049919 = sqrt(r16049915);
        double r16049920 = r16049918 - r16049919;
        return r16049920;
}
↓
double f(double x) {
        double r16049921 = 1.0;
        double r16049922 = x;
        double r16049923 = r16049922 + r16049921;
        double r16049924 = sqrt(r16049923);
        double r16049925 = sqrt(r16049922);
        double r16049926 = r16049924 + r16049925;
        double r16049927 = r16049921 / r16049926;
        return r16049927;
}

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. Simplified28.7

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

  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 2019158 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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