Average Error: 29.7 → 0.2
Time: 1.4m
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 r3596684 = x;
        double r3596685 = 1.0;
        double r3596686 = r3596684 + r3596685;
        double r3596687 = sqrt(r3596686);
        double r3596688 = sqrt(r3596684);
        double r3596689 = r3596687 - r3596688;
        return r3596689;
}
↓
double f(double x) {
        double r3596690 = 1.0;
        double r3596691 = x;
        double r3596692 = r3596691 + r3596690;
        double r3596693 = sqrt(r3596692);
        double r3596694 = sqrt(r3596691);
        double r3596695 = r3596693 + r3596694;
        double r3596696 = r3596690 / r3596695;
        return r3596696;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--29.5

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

  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 2019142 
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))