Average Error: 29.6 → 0.2
Time: 4.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 \cdot 1 - 0}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 \cdot 1 - 0}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}
double f(double x) {
        double r135112 = x;
        double r135113 = 1.0;
        double r135114 = r135112 + r135113;
        double r135115 = sqrt(r135114);
        double r135116 = sqrt(r135112);
        double r135117 = r135115 - r135116;
        return r135117;
}
↓
double f(double x) {
        double r135118 = 1.0;
        double r135119 = r135118 * r135118;
        double r135120 = 0.0;
        double r135121 = r135119 - r135120;
        double r135122 = x;
        double r135123 = r135122 + r135118;
        double r135124 = sqrt(r135123);
        double r135125 = sqrt(r135122);
        double r135126 = r135124 + r135125;
        double r135127 = r135118 * r135126;
        double r135128 = r135121 / r135127;
        return r135128;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 29.6

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

  3. Applied flip--29.4

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

  4. Simplified0.2

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

  6. Applied flip-+0.2

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

  7. Applied associate-/l/0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019346 +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)))