Average Error: 29.5 → 0.2
Time: 13.6s
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 r26127824 = x;
        double r26127825 = 1.0;
        double r26127826 = r26127824 + r26127825;
        double r26127827 = sqrt(r26127826);
        double r26127828 = sqrt(r26127824);
        double r26127829 = r26127827 - r26127828;
        return r26127829;
}
↓
double f(double x) {
        double r26127830 = 1.0;
        double r26127831 = x;
        double r26127832 = r26127831 + r26127830;
        double r26127833 = sqrt(r26127832);
        double r26127834 = sqrt(r26127831);
        double r26127835 = r26127833 + r26127834;
        double r26127836 = r26127830 / r26127835;
        return r26127836;
}

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

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.3

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

  4. Simplified28.9

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

  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 2019162 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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