Average Error: 20.2 → 0.4
Time: 18.6s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\frac{1}{x}}{\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}} \cdot \frac{1}{\frac{x + 1}{1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{\frac{1}{x}}{\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}} \cdot \frac{1}{\frac{x + 1}{1}}
double f(double x) {
        double r5860288 = 1.0;
        double r5860289 = x;
        double r5860290 = sqrt(r5860289);
        double r5860291 = r5860288 / r5860290;
        double r5860292 = r5860289 + r5860288;
        double r5860293 = sqrt(r5860292);
        double r5860294 = r5860288 / r5860293;
        double r5860295 = r5860291 - r5860294;
        return r5860295;
}
↓
double f(double x) {
        double r5860296 = 1.0;
        double r5860297 = x;
        double r5860298 = r5860296 / r5860297;
        double r5860299 = r5860297 + r5860296;
        double r5860300 = sqrt(r5860299);
        double r5860301 = r5860296 / r5860300;
        double r5860302 = sqrt(r5860297);
        double r5860303 = r5860296 / r5860302;
        double r5860304 = r5860301 + r5860303;
        double r5860305 = r5860298 / r5860304;
        double r5860306 = r5860299 / r5860296;
        double r5860307 = r5860296 / r5860306;
        double r5860308 = r5860305 * r5860307;
        return r5860308;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.2
Target0.7
Herbie0.4
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 20.2

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

  3. Applied flip--20.2

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

  5. Applied frac-times25.4

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

  6. Applied frac-times20.3

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

  7. Applied frac-sub20.0

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

  8. Simplified19.6

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

  9. Simplified19.6

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

  10. Taylor expanded around 0 5.6

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

  12. Applied *-un-lft-identity5.6

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

  13. Applied times-frac5.1

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

  14. Applied times-frac0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 
(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)))))