Average Error: 30.6 → 0.2
Time: 13.1s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{x + 1}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r617159 = x;
        double r617160 = 1.0;
        double r617161 = r617159 + r617160;
        double r617162 = sqrt(r617161);
        double r617163 = sqrt(r617159);
        double r617164 = r617162 - r617163;
        return r617164;
}
↓
double f(double x) {
        double r617165 = 1.0;
        double r617166 = x;
        double r617167 = sqrt(r617166);
        double r617168 = r617166 + r617165;
        double r617169 = sqrt(r617168);
        double r617170 = r617167 + r617169;
        double r617171 = r617165 / r617170;
        return r617171;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original30.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.6

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

  3. Applied flip--30.4

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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