Average Error: 10.0 → 0.3
Time: 41.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -97.68596649479380289449181873351335525513 \lor \neg \left(x \le 108.8580563138250454358058050274848937988\right):\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\sqrt{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x - 1}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -97.68596649479380289449181873351335525513 \lor \neg \left(x \le 108.8580563138250454358058050274848937988\right):\\
\;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\sqrt{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x - 1}}\\

\end{array}
double f(double x) {
        double r244153 = 1.0;
        double r244154 = x;
        double r244155 = r244154 + r244153;
        double r244156 = r244153 / r244155;
        double r244157 = 2.0;
        double r244158 = r244157 / r244154;
        double r244159 = r244156 - r244158;
        double r244160 = r244154 - r244153;
        double r244161 = r244153 / r244160;
        double r244162 = r244159 + r244161;
        return r244162;
}
↓
double f(double x) {
        double r244163 = x;
        double r244164 = -97.6859664947938;
        bool r244165 = r244163 <= r244164;
        double r244166 = 108.85805631382505;
        bool r244167 = r244163 <= r244166;
        double r244168 = !r244167;
        bool r244169 = r244165 || r244168;
        double r244170 = 2.0;
        double r244171 = 3.0;
        double r244172 = pow(r244163, r244171);
        double r244173 = r244170 / r244172;
        double r244174 = 7.0;
        double r244175 = pow(r244163, r244174);
        double r244176 = r244170 / r244175;
        double r244177 = 5.0;
        double r244178 = pow(r244163, r244177);
        double r244179 = r244170 / r244178;
        double r244180 = r244176 + r244179;
        double r244181 = r244173 + r244180;
        double r244182 = 1.0;
        double r244183 = r244163 + r244182;
        double r244184 = r244182 / r244183;
        double r244185 = r244170 / r244163;
        double r244186 = r244184 - r244185;
        double r244187 = sqrt(r244182);
        double r244188 = r244163 - r244182;
        double r244189 = cbrt(r244188);
        double r244190 = r244189 * r244189;
        double r244191 = r244187 / r244190;
        double r244192 = r244187 / r244189;
        double r244193 = r244191 * r244192;
        double r244194 = r244186 + r244193;
        double r244195 = r244169 ? r244181 : r244194;
        return r244195;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -97.6859664947938 or 108.85805631382505 < x

    1. Initial program 20.2

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)}\]

    if -97.6859664947938 < x < 108.85805631382505

    1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.1

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\color{blue}{\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}}}\]

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}\right) \cdot \sqrt[3]{x - 1}}\]

    5. Applied times-frac0.1

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x - 1}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -97.68596649479380289449181873351335525513 \lor \neg \left(x \le 108.8580563138250454358058050274848937988\right):\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\sqrt{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{\sqrt{1}}{\sqrt[3]{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))