Average Error: 9.9 → 0.1
Time: 35.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -111.6420365632021:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 108.84391371430509:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -111.6420365632021:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 108.84391371430509:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\

\end{array}
double f(double x) {
        double r2175460 = 1.0;
        double r2175461 = x;
        double r2175462 = r2175461 + r2175460;
        double r2175463 = r2175460 / r2175462;
        double r2175464 = 2.0;
        double r2175465 = r2175464 / r2175461;
        double r2175466 = r2175463 - r2175465;
        double r2175467 = r2175461 - r2175460;
        double r2175468 = r2175460 / r2175467;
        double r2175469 = r2175466 + r2175468;
        return r2175469;
}
↓
double f(double x) {
        double r2175470 = x;
        double r2175471 = -111.6420365632021;
        bool r2175472 = r2175470 <= r2175471;
        double r2175473 = 2.0;
        double r2175474 = 7.0;
        double r2175475 = pow(r2175470, r2175474);
        double r2175476 = r2175473 / r2175475;
        double r2175477 = r2175473 / r2175470;
        double r2175478 = r2175477 / r2175470;
        double r2175479 = r2175478 / r2175470;
        double r2175480 = 5.0;
        double r2175481 = pow(r2175470, r2175480);
        double r2175482 = r2175473 / r2175481;
        double r2175483 = r2175479 + r2175482;
        double r2175484 = r2175476 + r2175483;
        double r2175485 = 108.84391371430509;
        bool r2175486 = r2175470 <= r2175485;
        double r2175487 = 1.0;
        double r2175488 = r2175487 + r2175470;
        double r2175489 = r2175487 / r2175488;
        double r2175490 = r2175489 - r2175477;
        double r2175491 = r2175470 - r2175487;
        double r2175492 = r2175487 / r2175491;
        double r2175493 = r2175490 + r2175492;
        double r2175494 = r2175470 * r2175470;
        double r2175495 = r2175477 / r2175494;
        double r2175496 = r2175495 + r2175482;
        double r2175497 = r2175496 + r2175476;
        double r2175498 = r2175486 ? r2175493 : r2175497;
        double r2175499 = r2175472 ? r2175484 : r2175498;
        return r2175499;
}

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 3 regimes

  2. if x < -111.6420365632021

    1. Initial program 19.6

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    5. Applied *-un-lft-identity0.6

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

    6. Applied times-frac0.1

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

    8. Applied associate-*l/0.1

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

    9. Simplified0.1

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

    if -111.6420365632021 < x < 108.84391371430509

    1. Initial program 0.0

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

    if 108.84391371430509 < x

    1. Initial program 19.4

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    5. Applied *-un-lft-identity0.6

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

    6. Applied times-frac0.1

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

    8. Applied associate-*r/0.1

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

    9. Simplified0.1

      \[\leadsto \frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\color{blue}{\frac{2}{x}}}{x \cdot x}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -111.6420365632021:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{\frac{\frac{2}{x}}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 108.84391371430509:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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