Average Error: 30.3 → 0.2
Time: 17.7s
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 r18416597 = x;
        double r18416598 = 1.0;
        double r18416599 = r18416597 + r18416598;
        double r18416600 = sqrt(r18416599);
        double r18416601 = sqrt(r18416597);
        double r18416602 = r18416600 - r18416601;
        return r18416602;
}
↓
double f(double x) {
        double r18416603 = 1.0;
        double r18416604 = x;
        double r18416605 = r18416604 + r18416603;
        double r18416606 = sqrt(r18416605);
        double r18416607 = sqrt(r18416604);
        double r18416608 = r18416606 + r18416607;
        double r18416609 = r18416603 / r18416608;
        return r18416609;
}

Error

Bits error versus x

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Results

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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

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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