Average Error: 30.1 → 0.2
Time: 41.1s
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 r25285278 = x;
        double r25285279 = 1.0;
        double r25285280 = r25285278 + r25285279;
        double r25285281 = sqrt(r25285280);
        double r25285282 = sqrt(r25285278);
        double r25285283 = r25285281 - r25285282;
        return r25285283;
}
↓
double f(double x) {
        double r25285284 = 1.0;
        double r25285285 = x;
        double r25285286 = r25285285 + r25285284;
        double r25285287 = sqrt(r25285286);
        double r25285288 = sqrt(r25285285);
        double r25285289 = r25285287 + r25285288;
        double r25285290 = r25285284 / r25285289;
        return r25285290;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.1

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

  3. Applied flip--29.9

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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