Average Error: 30.3 → 0.2
Time: 15.1s
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 r323217 = x;
        double r323218 = 1.0;
        double r323219 = r323217 + r323218;
        double r323220 = sqrt(r323219);
        double r323221 = sqrt(r323217);
        double r323222 = r323220 - r323221;
        return r323222;
}
↓
double f(double x) {
        double r323223 = 1.0;
        double r323224 = 0.0;
        double r323225 = r323223 + r323224;
        double r323226 = x;
        double r323227 = r323226 + r323223;
        double r323228 = sqrt(r323227);
        double r323229 = sqrt(r323226);
        double r323230 = r323228 + r323229;
        double r323231 = r323225 / r323230;
        return r323231;
}

Error

Bits error versus x

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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--30.1

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

  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 2019291 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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