Average Error: 29.9 → 0.2
Time: 3.7s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1 + 0}{1 \cdot \sqrt{x + 1} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1 + 0}{1 \cdot \sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r595665 = x;
        double r595666 = 1.0;
        double r595667 = r595665 + r595666;
        double r595668 = sqrt(r595667);
        double r595669 = sqrt(r595665);
        double r595670 = r595668 - r595669;
        return r595670;
}
↓
double f(double x) {
        double r595671 = 1.0;
        double r595672 = 0.0;
        double r595673 = r595671 + r595672;
        double r595674 = 1.0;
        double r595675 = x;
        double r595676 = r595675 + r595671;
        double r595677 = sqrt(r595676);
        double r595678 = r595674 * r595677;
        double r595679 = sqrt(r595675);
        double r595680 = r595678 + r595679;
        double r595681 = r595673 / r595680;
        return r595681;
}

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 *-un-lft-identity0.2

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

  7. Applied sqrt-prod0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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