Average Error: 19.8 → 0.5
Time: 7.2s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot \frac{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot \frac{\frac{1}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r171545 = 1.0;
        double r171546 = x;
        double r171547 = sqrt(r171546);
        double r171548 = r171545 / r171547;
        double r171549 = r171546 + r171545;
        double r171550 = sqrt(r171549);
        double r171551 = r171545 / r171550;
        double r171552 = r171548 - r171551;
        return r171552;
}
↓
double f(double x) {
        double r171553 = 1.0;
        double r171554 = x;
        double r171555 = r171554 + r171553;
        double r171556 = sqrt(r171555);
        double r171557 = sqrt(r171554);
        double r171558 = r171556 + r171557;
        double r171559 = sqrt(r171558);
        double r171560 = r171553 / r171559;
        double r171561 = r171560 / r171559;
        double r171562 = r171553 * r171561;
        double r171563 = r171557 * r171556;
        double r171564 = r171562 / r171563;
        return r171564;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.7
Herbie0.5
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

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

  3. Applied frac-sub19.8

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

  4. Simplified19.8

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

  6. Applied flip--19.6

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

  7. Simplified19.2

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

  8. Taylor expanded around 0 0.4

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied associate-/r*0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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