Average Error: 20.1 → 0.4
Time: 16.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \sqrt{x}}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \sqrt{x}}}{\sqrt{x + 1}}
double f(double x) {
        double r1964525 = 1.0;
        double r1964526 = x;
        double r1964527 = sqrt(r1964526);
        double r1964528 = r1964525 / r1964527;
        double r1964529 = r1964526 + r1964525;
        double r1964530 = sqrt(r1964529);
        double r1964531 = r1964525 / r1964530;
        double r1964532 = r1964528 - r1964531;
        return r1964532;
}
↓
double f(double x) {
        double r1964533 = 1.0;
        double r1964534 = x;
        double r1964535 = r1964534 + r1964533;
        double r1964536 = sqrt(r1964535);
        double r1964537 = sqrt(r1964534);
        double r1964538 = r1964536 + r1964537;
        double r1964539 = r1964538 * r1964537;
        double r1964540 = r1964533 / r1964539;
        double r1964541 = r1964540 / r1964536;
        return r1964541;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.1

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

  3. Applied frac-sub20.0

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

  4. Simplified20.0

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

  6. Applied flip--19.8

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

  7. Simplified19.4

    \[\leadsto \frac{\frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  8. Using strategy rm

  9. Applied associate-/r*19.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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