Average Error: 9.6 → 0.3
Time: 15.5s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -94.31796734081924:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 89.72182339181649:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}, -2 \cdot \frac{1}{x}\right) + \left(\frac{1}{x - 1} + \mathsf{fma}\left(-\frac{1}{x}, 2, 2 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -94.31796734081924:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 89.72182339181649:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}, -2 \cdot \frac{1}{x}\right) + \left(\frac{1}{x - 1} + \mathsf{fma}\left(-\frac{1}{x}, 2, 2 \cdot \frac{1}{x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\

\end{array}
double f(double x) {
        double r2076404 = 1.0;
        double r2076405 = x;
        double r2076406 = r2076405 + r2076404;
        double r2076407 = r2076404 / r2076406;
        double r2076408 = 2.0;
        double r2076409 = r2076408 / r2076405;
        double r2076410 = r2076407 - r2076409;
        double r2076411 = r2076405 - r2076404;
        double r2076412 = r2076404 / r2076411;
        double r2076413 = r2076410 + r2076412;
        return r2076413;
}
↓
double f(double x) {
        double r2076414 = x;
        double r2076415 = -94.31796734081924;
        bool r2076416 = r2076414 <= r2076415;
        double r2076417 = 2.0;
        double r2076418 = 7.0;
        double r2076419 = pow(r2076414, r2076418);
        double r2076420 = r2076417 / r2076419;
        double r2076421 = 3.0;
        double r2076422 = pow(r2076414, r2076421);
        double r2076423 = r2076417 / r2076422;
        double r2076424 = 5.0;
        double r2076425 = pow(r2076414, r2076424);
        double r2076426 = r2076417 / r2076425;
        double r2076427 = r2076423 + r2076426;
        double r2076428 = r2076420 + r2076427;
        double r2076429 = 89.72182339181649;
        bool r2076430 = r2076414 <= r2076429;
        double r2076431 = 1.0;
        double r2076432 = r2076431 + r2076414;
        double r2076433 = cbrt(r2076432);
        double r2076434 = r2076431 / r2076433;
        double r2076435 = r2076433 * r2076433;
        double r2076436 = r2076431 / r2076435;
        double r2076437 = r2076434 * r2076436;
        double r2076438 = -2.0;
        double r2076439 = r2076431 / r2076414;
        double r2076440 = r2076438 * r2076439;
        double r2076441 = fma(r2076431, r2076437, r2076440);
        double r2076442 = r2076414 - r2076431;
        double r2076443 = r2076431 / r2076442;
        double r2076444 = -r2076439;
        double r2076445 = r2076417 * r2076439;
        double r2076446 = fma(r2076444, r2076417, r2076445);
        double r2076447 = r2076443 + r2076446;
        double r2076448 = r2076441 + r2076447;
        double r2076449 = r2076430 ? r2076448 : r2076428;
        double r2076450 = r2076416 ? r2076428 : r2076449;
        return r2076450;
}

Error

Bits error versus x

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -94.31796734081924 or 89.72182339181649 < x

    1. Initial program 19.5

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    5. Applied pow10.5

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

    6. Applied pow10.5

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

    7. Applied pow-prod-up0.5

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

    8. Applied pow10.5

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

    9. Applied pow-prod-up0.5

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

    10. Simplified0.5

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

    if -94.31796734081924 < x < 89.72182339181649

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied prod-diff0.0

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

    6. Applied associate-+l+0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{x + 1}, -\frac{1}{x} \cdot 2\right) + \left(\mathsf{fma}\left(-\frac{1}{x}, 2, \frac{1}{x} \cdot 2\right) + \frac{1}{x - 1}\right)}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.0

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

    9. Applied add-sqr-sqrt0.0

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

    10. Applied times-frac0.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -94.31796734081924:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 89.72182339181649:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{\sqrt[3]{1 + x}} \cdot \frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}, -2 \cdot \frac{1}{x}\right) + \left(\frac{1}{x - 1} + \mathsf{fma}\left(-\frac{1}{x}, 2, 2 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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))))