Average Error: 30.1 → 0.2
Time: 52.0s
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 r6859767 = x;
        double r6859768 = 1.0;
        double r6859769 = r6859767 + r6859768;
        double r6859770 = sqrt(r6859769);
        double r6859771 = sqrt(r6859767);
        double r6859772 = r6859770 - r6859771;
        return r6859772;
}
↓
double f(double x) {
        double r6859773 = 1.0;
        double r6859774 = x;
        double r6859775 = r6859774 + r6859773;
        double r6859776 = sqrt(r6859775);
        double r6859777 = sqrt(r6859774);
        double r6859778 = r6859776 + r6859777;
        double r6859779 = r6859773 / r6859778;
        return r6859779;
}

Error

Bits error versus x

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Your Program's Arguments

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. Taylor expanded around inf 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 2019107 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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