Average Error: 30.0 → 0.2
Time: 13.6s
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 r387822 = x;
        double r387823 = 1.0;
        double r387824 = r387822 + r387823;
        double r387825 = sqrt(r387824);
        double r387826 = sqrt(r387822);
        double r387827 = r387825 - r387826;
        return r387827;
}
↓
double f(double x) {
        double r387828 = 1.0;
        double r387829 = x;
        double r387830 = r387829 + r387828;
        double r387831 = sqrt(r387830);
        double r387832 = sqrt(r387829);
        double r387833 = r387831 + r387832;
        double r387834 = r387828 / r387833;
        return r387834;
}

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. Simplified30.0

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}}\]
  3. Using strategy rm

  4. Applied flip--29.9

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

  5. Simplified0.2

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

  6. 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)))