Average Error: 19.9 → 0.4
Time: 42.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[1 \cdot \frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
1 \cdot \frac{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x}}}{1 \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}
double f(double x) {
        double r4103809 = 1.0;
        double r4103810 = x;
        double r4103811 = sqrt(r4103810);
        double r4103812 = r4103809 / r4103811;
        double r4103813 = r4103810 + r4103809;
        double r4103814 = sqrt(r4103813);
        double r4103815 = r4103809 / r4103814;
        double r4103816 = r4103812 - r4103815;
        return r4103816;
}
↓
double f(double x) {
        double r4103817 = 1.0;
        double r4103818 = 1.0;
        double r4103819 = x;
        double r4103820 = r4103819 + r4103817;
        double r4103821 = sqrt(r4103820);
        double r4103822 = r4103818 / r4103821;
        double r4103823 = sqrt(r4103819);
        double r4103824 = r4103822 / r4103823;
        double r4103825 = r4103823 + r4103821;
        double r4103826 = r4103817 * r4103825;
        double r4103827 = r4103824 / r4103826;
        double r4103828 = r4103817 * r4103827;
        return r4103828;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.9
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.9

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

  3. Applied frac-sub19.9

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

  5. Applied flip--19.7

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

  6. Applied associate-/l/19.7

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

  7. Taylor expanded around 0 0.8

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

  9. Applied div-inv0.8

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +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)))))