Average Error: 20.2 → 0.4
Time: 22.9s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\frac{x}{1}} \cdot \frac{\frac{1}{x + 1}}{\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\frac{x}{1}} \cdot \frac{\frac{1}{x + 1}}{\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}}
double f(double x) {
        double r6639319 = 1.0;
        double r6639320 = x;
        double r6639321 = sqrt(r6639320);
        double r6639322 = r6639319 / r6639321;
        double r6639323 = r6639320 + r6639319;
        double r6639324 = sqrt(r6639323);
        double r6639325 = r6639319 / r6639324;
        double r6639326 = r6639322 - r6639325;
        return r6639326;
}
↓
double f(double x) {
        double r6639327 = 1.0;
        double r6639328 = x;
        double r6639329 = r6639328 / r6639327;
        double r6639330 = r6639327 / r6639329;
        double r6639331 = r6639328 + r6639327;
        double r6639332 = r6639327 / r6639331;
        double r6639333 = sqrt(r6639331);
        double r6639334 = r6639327 / r6639333;
        double r6639335 = sqrt(r6639328);
        double r6639336 = r6639327 / r6639335;
        double r6639337 = r6639334 + r6639336;
        double r6639338 = r6639332 / r6639337;
        double r6639339 = r6639330 * r6639338;
        return r6639339;
}

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

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

  10. Taylor expanded around 0 5.6

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

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

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

  13. Applied times-frac5.1

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

  14. Applied times-frac0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 +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)))))