Average Error: 29.3 → 0.2
Time: 19.0s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{x} + \sqrt{x + 1}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{x} + \sqrt{x + 1}}
double f(double x) {
        double r4781204 = x;
        double r4781205 = 1.0;
        double r4781206 = r4781204 + r4781205;
        double r4781207 = sqrt(r4781206);
        double r4781208 = sqrt(r4781204);
        double r4781209 = r4781207 - r4781208;
        return r4781209;
}
↓
double f(double x) {
        double r4781210 = 1.0;
        double r4781211 = x;
        double r4781212 = sqrt(r4781211);
        double r4781213 = r4781211 + r4781210;
        double r4781214 = sqrt(r4781213);
        double r4781215 = r4781212 + r4781214;
        double r4781216 = r4781210 / r4781215;
        return r4781216;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.3

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

  3. Applied flip--29.1

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

  4. Simplified28.7

    \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

  5. Simplified28.7

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity28.7

    \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{1 \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}}\]

  8. Applied associate-/r*28.7

    \[\leadsto \color{blue}{\frac{\frac{\left(1 + x\right) - x}{1}}{\sqrt{x} + \sqrt{1 + x}}}\]

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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