Average Error: 19.6 → 0.7
Time: 11.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1 \cdot 1}{\left(x + 1\right) \cdot \sqrt{x} + \sqrt{x + 1} \cdot x}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1 \cdot 1}{\left(x + 1\right) \cdot \sqrt{x} + \sqrt{x + 1} \cdot x}
double f(double x) {
        double r189954 = 1.0;
        double r189955 = x;
        double r189956 = sqrt(r189955);
        double r189957 = r189954 / r189956;
        double r189958 = r189955 + r189954;
        double r189959 = sqrt(r189958);
        double r189960 = r189954 / r189959;
        double r189961 = r189957 - r189960;
        return r189961;
}
↓
double f(double x) {
        double r189962 = 1.0;
        double r189963 = r189962 * r189962;
        double r189964 = x;
        double r189965 = r189964 + r189962;
        double r189966 = sqrt(r189964);
        double r189967 = r189965 * r189966;
        double r189968 = sqrt(r189965);
        double r189969 = r189968 * r189964;
        double r189970 = r189967 + r189969;
        double r189971 = r189963 / r189970;
        return r189971;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 19.6

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

  3. Applied frac-sub19.6

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

  4. Simplified19.6

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

  6. Applied flip--19.4

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

  7. Applied associate-*r/19.4

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

  8. Applied associate-/l/19.4

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

  9. Simplified19.4

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

  10. Taylor expanded around 0 0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2020046 +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)))))