Average Error: 30.3 → 0.2
Time: 17.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r25365250 = x;
        double r25365251 = 1.0;
        double r25365252 = r25365250 + r25365251;
        double r25365253 = sqrt(r25365252);
        double r25365254 = sqrt(r25365250);
        double r25365255 = r25365253 - r25365254;
        return r25365255;
}
↓
double f(double x) {
        double r25365256 = 1.0;
        double r25365257 = x;
        double r25365258 = r25365257 + r25365256;
        double r25365259 = sqrt(r25365258);
        double r25365260 = sqrt(r25365257);
        double r25365261 = r25365259 + r25365260;
        double r25365262 = r25365256 / r25365261;
        return r25365262;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.0

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

  4. Simplified29.6

    \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified29.6

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\]

  6. Taylor expanded around 0 0.2

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

  7. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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