Average Error: 30.3 → 0.2
Time: 19.9s
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 r5841176 = x;
        double r5841177 = 1.0;
        double r5841178 = r5841176 + r5841177;
        double r5841179 = sqrt(r5841178);
        double r5841180 = sqrt(r5841176);
        double r5841181 = r5841179 - r5841180;
        return r5841181;
}
↓
double f(double x) {
        double r5841182 = 1.0;
        double r5841183 = x;
        double r5841184 = r5841183 + r5841182;
        double r5841185 = sqrt(r5841184);
        double r5841186 = sqrt(r5841183);
        double r5841187 = r5841185 + r5841186;
        double r5841188 = r5841182 / r5841187;
        return r5841188;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.3

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

  3. Applied flip--30.1

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))