Average Error: 20.0 → 0.6
Time: 6.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot 1}{\mathsf{fma}\left(x + 1, \sqrt{x}, \sqrt{x + 1} \cdot x\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot 1}{\mathsf{fma}\left(x + 1, \sqrt{x}, \sqrt{x + 1} \cdot x\right)}
double f(double x) {
        double r136652 = 1.0;
        double r136653 = x;
        double r136654 = sqrt(r136653);
        double r136655 = r136652 / r136654;
        double r136656 = r136653 + r136652;
        double r136657 = sqrt(r136656);
        double r136658 = r136652 / r136657;
        double r136659 = r136655 - r136658;
        return r136659;
}
↓
double f(double x) {
        double r136660 = 1.0;
        double r136661 = r136660 * r136660;
        double r136662 = x;
        double r136663 = r136662 + r136660;
        double r136664 = sqrt(r136662);
        double r136665 = sqrt(r136663);
        double r136666 = r136665 * r136662;
        double r136667 = fma(r136663, r136664, r136666);
        double r136668 = r136661 / r136667;
        return r136668;
}

Error

Bits error versus x

Target

Original20.0
Target0.6
Herbie0.6
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.0

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

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

  6. Applied flip--19.8

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

  7. Simplified19.3

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

  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 associate-*r/0.4

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

  11. Applied associate-/l/0.8

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(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)))))