Average Error: 19.4 → 0.3
Time: 13.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}
double f(double x) {
        double r108942 = 1.0;
        double r108943 = x;
        double r108944 = sqrt(r108943);
        double r108945 = r108942 / r108944;
        double r108946 = r108943 + r108942;
        double r108947 = sqrt(r108946);
        double r108948 = r108942 / r108947;
        double r108949 = r108945 - r108948;
        return r108949;
}
↓
double f(double x) {
        double r108950 = 1.0;
        double r108951 = x;
        double r108952 = sqrt(r108951);
        double r108953 = r108950 / r108952;
        double r108954 = r108951 + r108950;
        double r108955 = sqrt(r108954);
        double r108956 = r108952 * r108955;
        double r108957 = r108954 + r108956;
        double r108958 = r108950 / r108957;
        double r108959 = r108953 * r108958;
        return r108959;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
Target0.8
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.4

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

  3. Applied frac-sub19.4

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

  4. Simplified19.4

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

  6. Applied flip--19.2

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

  7. Simplified18.9

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

  8. Simplified18.9

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

  9. Taylor expanded around 0 0.4

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

  11. Applied times-frac0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 
(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)))))