Average Error: 29.7 → 0.2
Time: 4.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\sqrt{1} \cdot \frac{\sqrt{1}}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r111030 = x;
        double r111031 = 1.0;
        double r111032 = r111030 + r111031;
        double r111033 = sqrt(r111032);
        double r111034 = sqrt(r111030);
        double r111035 = r111033 - r111034;
        return r111035;
}
↓
double f(double x) {
        double r111036 = 1.0;
        double r111037 = sqrt(r111036);
        double r111038 = x;
        double r111039 = r111038 + r111036;
        double r111040 = sqrt(r111039);
        double r111041 = sqrt(r111038);
        double r111042 = r111040 + r111041;
        double r111043 = r111037 / r111042;
        double r111044 = r111037 * r111043;
        return r111044;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.7

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

  3. Applied flip--29.5

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

  4. Simplified0.2

    \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.2

    \[\leadsto \frac{1 + 0}{\color{blue}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  7. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{\color{blue}{\sqrt{1 + 0} \cdot \sqrt{1 + 0}}}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]

  8. Applied times-frac0.2

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

  9. Simplified0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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