Average Error: 39.7 → 11.5
Time: 18.5s
Precision: 64
\[\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}\]
\[\frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}\]
\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}
\frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}
double f(double x) {
        double r57087 = x;
        double r57088 = 1.0;
        double r57089 = r57087 - r57088;
        double r57090 = sqrt(r57089);
        double r57091 = r57087 + r57088;
        double r57092 = r57090 / r57091;
        double r57093 = sqrt(r57091);
        double r57094 = r57093 / r57089;
        double r57095 = r57092 - r57094;
        return r57095;
}

double f(double x) {
        double r57096 = 14.0;
        double r57097 = x;
        double r57098 = 4.0;
        double r57099 = pow(r57097, r57098);
        double r57100 = r57096 / r57099;
        double r57101 = 22.0;
        double r57102 = 6.0;
        double r57103 = pow(r57097, r57102);
        double r57104 = r57101 / r57103;
        double r57105 = r57100 + r57104;
        double r57106 = 6.0;
        double r57107 = r57097 * r57097;
        double r57108 = r57106 / r57107;
        double r57109 = r57105 + r57108;
        double r57110 = -r57109;
        double r57111 = 1.0;
        double r57112 = r57097 - r57111;
        double r57113 = sqrt(r57112);
        double r57114 = r57097 + r57111;
        double r57115 = r57113 / r57114;
        double r57116 = sqrt(r57114);
        double r57117 = r57116 / r57112;
        double r57118 = r57115 + r57117;
        double r57119 = r57110 / r57118;
        return r57119;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.7

    \[\frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1}\]
  2. Using strategy rm
  3. Applied flip--39.7

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{x - 1}}{x + 1} \cdot \frac{\sqrt{x - 1}}{x + 1} - \frac{\sqrt{x + 1}}{x - 1} \cdot \frac{\sqrt{x + 1}}{x - 1}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}}\]
  4. Simplified39.7

    \[\leadsto \frac{\color{blue}{\frac{x - 1}{\left(x + 1\right) \cdot \left(x + 1\right)} - \frac{x + 1}{\left(x - 1\right) \cdot \left(x - 1\right)}}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}\]
  5. Taylor expanded around inf 11.5

    \[\leadsto \frac{\color{blue}{-\left(6 \cdot \frac{1}{{x}^{2}} + \left(14 \cdot \frac{1}{{x}^{4}} + 22 \cdot \frac{1}{{x}^{6}}\right)\right)}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}\]
  6. Simplified11.5

    \[\leadsto \frac{\color{blue}{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}\]
  7. Final simplification11.5

    \[\leadsto \frac{-\left(\left(\frac{14}{{x}^{4}} + \frac{22}{{x}^{6}}\right) + \frac{6}{x \cdot x}\right)}{\frac{\sqrt{x - 1}}{x + 1} + \frac{\sqrt{x + 1}}{x - 1}}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(x-1)/(x+1)-sqrt(x+1)/(x-1)"
  :precision binary64
  (- (/ (sqrt (- x 1)) (+ x 1)) (/ (sqrt (+ x 1)) (- x 1))))