Average Error: 29.8 → 0.2
Time: 30.7s
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 r8985722 = x;
        double r8985723 = 1.0;
        double r8985724 = r8985722 + r8985723;
        double r8985725 = sqrt(r8985724);
        double r8985726 = sqrt(r8985722);
        double r8985727 = r8985725 - r8985726;
        return r8985727;
}
↓
double f(double x) {
        double r8985728 = 1.0;
        double r8985729 = x;
        double r8985730 = r8985729 + r8985728;
        double r8985731 = sqrt(r8985730);
        double r8985732 = sqrt(r8985729);
        double r8985733 = r8985731 + r8985732;
        double r8985734 = r8985728 / r8985733;
        return r8985734;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.8

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

  3. Applied flip--29.6

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

  4. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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