Average Error: 30.1 → 0.2
Time: 20.0s
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 r20210750 = x;
        double r20210751 = 1.0;
        double r20210752 = r20210750 + r20210751;
        double r20210753 = sqrt(r20210752);
        double r20210754 = sqrt(r20210750);
        double r20210755 = r20210753 - r20210754;
        return r20210755;
}
↓
double f(double x) {
        double r20210756 = 1.0;
        double r20210757 = x;
        double r20210758 = r20210757 + r20210756;
        double r20210759 = sqrt(r20210758);
        double r20210760 = sqrt(r20210757);
        double r20210761 = r20210759 + r20210760;
        double r20210762 = r20210756 / r20210761;
        return r20210762;
}

Error

Bits error versus x

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Results

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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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 +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)))