Average Error: 29.3 → 0.2
Time: 16.6s
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 r25052418 = x;
        double r25052419 = 1.0;
        double r25052420 = r25052418 + r25052419;
        double r25052421 = sqrt(r25052420);
        double r25052422 = sqrt(r25052418);
        double r25052423 = r25052421 - r25052422;
        return r25052423;
}
↓
double f(double x) {
        double r25052424 = 1.0;
        double r25052425 = x;
        double r25052426 = r25052425 + r25052424;
        double r25052427 = sqrt(r25052426);
        double r25052428 = sqrt(r25052425);
        double r25052429 = r25052427 + r25052428;
        double r25052430 = r25052424 / r25052429;
        return r25052430;
}

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)))