Average Error: 9.9 → 0.1
Time: 15.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -17754.39869344371254555881023406982421875 \lor \neg \left(x \le 1913.506658456775085141998715698719024658\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -17754.39869344371254555881023406982421875 \lor \neg \left(x \le 1913.506658456775085141998715698719024658\right):\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\

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

\end{array}
double f(double x) {
        double r121916 = 1.0;
        double r121917 = x;
        double r121918 = r121917 + r121916;
        double r121919 = r121916 / r121918;
        double r121920 = 2.0;
        double r121921 = r121920 / r121917;
        double r121922 = r121919 - r121921;
        double r121923 = r121917 - r121916;
        double r121924 = r121916 / r121923;
        double r121925 = r121922 + r121924;
        return r121925;
}
↓
double f(double x) {
        double r121926 = x;
        double r121927 = -17754.398693443713;
        bool r121928 = r121926 <= r121927;
        double r121929 = 1913.506658456775;
        bool r121930 = r121926 <= r121929;
        double r121931 = !r121930;
        bool r121932 = r121928 || r121931;
        double r121933 = 2.0;
        double r121934 = 7.0;
        double r121935 = pow(r121926, r121934);
        double r121936 = r121933 / r121935;
        double r121937 = 5.0;
        double r121938 = pow(r121926, r121937);
        double r121939 = r121933 / r121938;
        double r121940 = r121933 / r121926;
        double r121941 = r121926 * r121926;
        double r121942 = r121940 / r121941;
        double r121943 = r121939 + r121942;
        double r121944 = r121936 + r121943;
        double r121945 = 1.0;
        double r121946 = r121945 * r121926;
        double r121947 = r121926 + r121945;
        double r121948 = r121947 * r121933;
        double r121949 = r121946 - r121948;
        double r121950 = r121926 - r121945;
        double r121951 = r121949 * r121950;
        double r121952 = r121947 * r121926;
        double r121953 = r121952 * r121945;
        double r121954 = r121951 + r121953;
        double r121955 = r121952 * r121950;
        double r121956 = r121954 / r121955;
        double r121957 = r121932 ? r121944 : r121956;
        return r121957;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -17754.398693443713 or 1913.506658456775 < x

    1. Initial program 19.6

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    5. Applied cube-mult0.5

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

    6. Applied associate-/r*0.1

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

    if -17754.398693443713 < x < 1913.506658456775

    1. Initial program 0.2

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

    3. Applied frac-sub0.2

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

    4. Applied frac-add0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -17754.39869344371254555881023406982421875 \lor \neg \left(x \le 1913.506658456775085141998715698719024658\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))