Average Error: 29.5 → 0.2
Time: 4.3s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r116089 = x;
        double r116090 = 1.0;
        double r116091 = r116089 + r116090;
        double r116092 = sqrt(r116091);
        double r116093 = sqrt(r116089);
        double r116094 = r116092 - r116093;
        return r116094;
}
↓
double f(double x) {
        double r116095 = 1.0;
        double r116096 = 0.0;
        double r116097 = r116095 + r116096;
        double r116098 = x;
        double r116099 = r116098 + r116095;
        double r116100 = sqrt(r116099);
        double r116101 = sqrt(r116098);
        double r116102 = r116100 + r116101;
        double r116103 = r116097 / r116102;
        return r116103;
}

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

Derivation

  1. Initial program 29.5

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

  3. Applied flip--29.3

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

    \[\leadsto \frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))