Average Error: 29.9 → 0.2
Time: 9.2s
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 r582225 = x;
        double r582226 = 1.0;
        double r582227 = r582225 + r582226;
        double r582228 = sqrt(r582227);
        double r582229 = sqrt(r582225);
        double r582230 = r582228 - r582229;
        return r582230;
}
↓
double f(double x) {
        double r582231 = 1.0;
        double r582232 = x;
        double r582233 = r582232 + r582231;
        double r582234 = sqrt(r582233);
        double r582235 = sqrt(r582232);
        double r582236 = r582234 + r582235;
        double r582237 = r582231 / r582236;
        return r582237;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.8

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 +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)))