Average Error: 29.7 → 0.2
Time: 10.8s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]
\sqrt{x + 1} - \sqrt{x}
↓
{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}
double f(double x) {
        double r1620204 = x;
        double r1620205 = 1.0;
        double r1620206 = r1620204 + r1620205;
        double r1620207 = sqrt(r1620206);
        double r1620208 = sqrt(r1620204);
        double r1620209 = r1620207 - r1620208;
        return r1620209;
}
↓
double f(double x) {
        double r1620210 = 1.0;
        double r1620211 = x;
        double r1620212 = r1620210 + r1620211;
        double r1620213 = sqrt(r1620212);
        double r1620214 = sqrt(r1620211);
        double r1620215 = r1620213 + r1620214;
        double r1620216 = r1620215 * r1620215;
        double r1620217 = -0.5;
        double r1620218 = pow(r1620216, r1620217);
        return r1620218;
}

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--29.4

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt0.3

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

  8. Applied pow1/20.3

    \[\leadsto \frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}} \cdot \color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}}\]

  9. Applied pow1/20.3

    \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}} \cdot {\left(\sqrt{x + 1} + \sqrt{x}\right)}^{\frac{1}{2}}}\]

  10. Applied pow-prod-down0.2

    \[\leadsto \frac{1}{\color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\frac{1}{2}}}}\]

  11. Applied pow-flip0.2

    \[\leadsto \color{blue}{{\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}}\]

  12. Simplified0.2

    \[\leadsto {\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\color{blue}{\frac{-1}{2}}}\]

  13. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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