Average Error: 19.7 → 0.4
Time: 17.1s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1} \cdot \frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot 1} \cdot \frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}
double f(double x) {
        double r131587 = 1.0;
        double r131588 = x;
        double r131589 = sqrt(r131588);
        double r131590 = r131587 / r131589;
        double r131591 = r131588 + r131587;
        double r131592 = sqrt(r131591);
        double r131593 = r131587 / r131592;
        double r131594 = r131590 - r131593;
        return r131594;
}
↓
double f(double x) {
        double r131595 = 1.0;
        double r131596 = x;
        double r131597 = r131596 + r131595;
        double r131598 = sqrt(r131597);
        double r131599 = sqrt(r131596);
        double r131600 = r131598 + r131599;
        double r131601 = r131600 * r131595;
        double r131602 = r131595 / r131601;
        double r131603 = 1.0;
        double r131604 = r131603 / r131598;
        double r131605 = r131604 / r131599;
        double r131606 = r131602 * r131605;
        return r131606;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.6
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied frac-sub19.7

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

  4. Simplified19.7

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

  6. Applied flip--19.5

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

  7. Simplified19.1

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

  8. Simplified19.1

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

  9. Taylor expanded around 0 0.4

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

  11. Applied div-inv0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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