Average Error: 19.7 → 0.3
Time: 2.7m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\left(\sqrt{x}\right), \left(\sqrt{x + 1}\right), x\right)}}{\sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{\mathsf{fma}\left(\left(\sqrt{x}\right), \left(\sqrt{x + 1}\right), x\right)}}{\sqrt{x + 1}}
double f(double x) {
        double r20353474 = 1.0;
        double r20353475 = x;
        double r20353476 = sqrt(r20353475);
        double r20353477 = r20353474 / r20353476;
        double r20353478 = r20353475 + r20353474;
        double r20353479 = sqrt(r20353478);
        double r20353480 = r20353474 / r20353479;
        double r20353481 = r20353477 - r20353480;
        return r20353481;
}
↓
double f(double x) {
        double r20353482 = 1.0;
        double r20353483 = x;
        double r20353484 = sqrt(r20353483);
        double r20353485 = r20353483 + r20353482;
        double r20353486 = sqrt(r20353485);
        double r20353487 = fma(r20353484, r20353486, r20353483);
        double r20353488 = r20353482 / r20353487;
        double r20353489 = r20353488 / r20353486;
        return r20353489;
}

Error

Bits error versus x

Target

Original19.7
Target0.6
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

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

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

  6. Applied flip--19.5

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

  7. Simplified0.4

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))))