Average Error: 30.0 → 0.2
Time: 16.0s
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 r442813 = x;
        double r442814 = 1.0;
        double r442815 = r442813 + r442814;
        double r442816 = sqrt(r442815);
        double r442817 = sqrt(r442813);
        double r442818 = r442816 - r442817;
        return r442818;
}
↓
double f(double x) {
        double r442819 = 1.0;
        double r442820 = x;
        double r442821 = r442820 + r442819;
        double r442822 = sqrt(r442821);
        double r442823 = sqrt(r442820);
        double r442824 = r442822 + r442823;
        double r442825 = r442819 / r442824;
        return r442825;
}

Error

Bits error versus x

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Results

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Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x}\]
  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 2019194 
(FPCore (x)
  :name "Main:bigenough3 from C"

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

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