Average Error: 30.0 → 0.3
Time: 13.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}
double f(double x) {
        double r92615 = x;
        double r92616 = 1.0;
        double r92617 = r92615 + r92616;
        double r92618 = sqrt(r92617);
        double r92619 = sqrt(r92615);
        double r92620 = r92618 - r92619;
        return r92620;
}
↓
double f(double x) {
        double r92621 = 1.0;
        double r92622 = x;
        double r92623 = r92622 + r92621;
        double r92624 = sqrt(r92623);
        double r92625 = sqrt(r92624);
        double r92626 = r92625 * r92625;
        double r92627 = sqrt(r92622);
        double r92628 = r92626 + r92627;
        double r92629 = r92621 / r92628;
        return r92629;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.3
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x}\]

  2. Simplified30.0

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

  4. Applied flip--29.9

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

  5. Simplified0.2

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

  7. Applied add-sqr-sqrt0.2

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

  8. Applied sqrt-prod0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))