Average Error: 29.9 → 0.2
Time: 4.2s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{\sqrt{1 + 0}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{1}}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{\sqrt{1 + 0}}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{1}}}
double f(double x) {
        double r110012 = x;
        double r110013 = 1.0;
        double r110014 = r110012 + r110013;
        double r110015 = sqrt(r110014);
        double r110016 = sqrt(r110012);
        double r110017 = r110015 - r110016;
        return r110017;
}
↓
double f(double x) {
        double r110018 = 1.0;
        double r110019 = 0.0;
        double r110020 = r110018 + r110019;
        double r110021 = sqrt(r110020);
        double r110022 = x;
        double r110023 = r110022 + r110018;
        double r110024 = sqrt(r110023);
        double r110025 = sqrt(r110022);
        double r110026 = r110024 + r110025;
        double r110027 = sqrt(r110018);
        double r110028 = r110026 / r110027;
        double r110029 = r110021 / r110028;
        return r110029;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.9

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

  3. Applied flip--29.7

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied associate-/l*0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 +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)))