Average Error: 30.1 → 0.2
Time: 15.5s
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 r3262165 = x;
        double r3262166 = 1.0;
        double r3262167 = r3262165 + r3262166;
        double r3262168 = sqrt(r3262167);
        double r3262169 = sqrt(r3262165);
        double r3262170 = r3262168 - r3262169;
        return r3262170;
}
↓
double f(double x) {
        double r3262171 = 1.0;
        double r3262172 = x;
        double r3262173 = r3262172 + r3262171;
        double r3262174 = sqrt(r3262173);
        double r3262175 = sqrt(r3262172);
        double r3262176 = r3262174 + r3262175;
        double r3262177 = r3262171 / r3262176;
        return r3262177;
}

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. 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 2019133 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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