Average Error: 30.1 → 0.2
Time: 4.5s
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 r497945 = x;
        double r497946 = 1.0;
        double r497947 = r497945 + r497946;
        double r497948 = sqrt(r497947);
        double r497949 = sqrt(r497945);
        double r497950 = r497948 - r497949;
        return r497950;
}
↓
double f(double x) {
        double r497951 = 1.0;
        double r497952 = 0.0;
        double r497953 = r497951 + r497952;
        double r497954 = x;
        double r497955 = r497954 + r497951;
        double r497956 = sqrt(r497955);
        double r497957 = sqrt(r497954);
        double r497958 = r497956 + r497957;
        double r497959 = r497953 / r497958;
        return r497959;
}

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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020021 +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)))