Average Error: 19.3 → 0.3
Time: 7.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x + 1\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x + 1\right)}
double f(double x) {
        double r195598 = 1.0;
        double r195599 = x;
        double r195600 = sqrt(r195599);
        double r195601 = r195598 / r195600;
        double r195602 = r195599 + r195598;
        double r195603 = sqrt(r195602);
        double r195604 = r195598 / r195603;
        double r195605 = r195601 - r195604;
        return r195605;
}
↓
double f(double x) {
        double r195606 = 1.0;
        double r195607 = x;
        double r195608 = sqrt(r195607);
        double r195609 = r195606 / r195608;
        double r195610 = r195607 + r195606;
        double r195611 = sqrt(r195610);
        double r195612 = fma(r195611, r195608, r195610);
        double r195613 = r195606 / r195612;
        double r195614 = r195609 * r195613;
        return r195614;
}

Error

Bits error versus x

Target

Original19.3
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.3

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

  3. Applied frac-sub19.2

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

  4. Simplified19.2

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

  6. Applied flip--19.1

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

  7. Simplified18.7

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

  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 times-frac0.4

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020033 +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)))))