Average Error: 9.5 → 0.0
Time: 8.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -136.1881966537822 \lor \neg \left(x \le 119.57592943098413\right):\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, 2 \cdot {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, \frac{x - 1}{\sqrt[3]{1}}, \left(\left(x + 1\right) \cdot x\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \frac{x - 1}{\sqrt[3]{1}}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -136.1881966537822 \lor \neg \left(x \le 119.57592943098413\right):\\
\;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, 2 \cdot {x}^{\left(-3\right)}\right)\right)\\

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

\end{array}
double f(double x) {
        double r151063 = 1.0;
        double r151064 = x;
        double r151065 = r151064 + r151063;
        double r151066 = r151063 / r151065;
        double r151067 = 2.0;
        double r151068 = r151067 / r151064;
        double r151069 = r151066 - r151068;
        double r151070 = r151064 - r151063;
        double r151071 = r151063 / r151070;
        double r151072 = r151069 + r151071;
        return r151072;
}
↓
double f(double x) {
        double r151073 = x;
        double r151074 = -136.1881966537822;
        bool r151075 = r151073 <= r151074;
        double r151076 = 119.57592943098413;
        bool r151077 = r151073 <= r151076;
        double r151078 = !r151077;
        bool r151079 = r151075 || r151078;
        double r151080 = 2.0;
        double r151081 = 1.0;
        double r151082 = 7.0;
        double r151083 = pow(r151073, r151082);
        double r151084 = r151081 / r151083;
        double r151085 = 5.0;
        double r151086 = pow(r151073, r151085);
        double r151087 = r151081 / r151086;
        double r151088 = 3.0;
        double r151089 = -r151088;
        double r151090 = pow(r151073, r151089);
        double r151091 = r151080 * r151090;
        double r151092 = fma(r151080, r151087, r151091);
        double r151093 = fma(r151080, r151084, r151092);
        double r151094 = 1.0;
        double r151095 = r151094 * r151073;
        double r151096 = r151073 + r151094;
        double r151097 = r151096 * r151080;
        double r151098 = r151095 - r151097;
        double r151099 = r151073 - r151094;
        double r151100 = cbrt(r151094);
        double r151101 = r151099 / r151100;
        double r151102 = r151096 * r151073;
        double r151103 = r151100 * r151100;
        double r151104 = r151102 * r151103;
        double r151105 = fma(r151098, r151101, r151104);
        double r151106 = r151102 * r151101;
        double r151107 = r151105 / r151106;
        double r151108 = r151079 ? r151093 : r151107;
        return r151108;
}

Error

Bits error versus x

Target

Original9.5
Target0.3
Herbie0.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -136.1881966537822 or 119.57592943098413 < x

    1. Initial program 18.9

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

    3. Applied add-cube-cbrt18.9

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

    4. Applied associate-/l*18.9

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

    5. Taylor expanded around inf 0.6

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

    6. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, 2 \cdot \frac{1}{{x}^{3}}\right)\right)}\]
    7. Using strategy rm

    8. Applied pow-flip0.0

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

    if -136.1881966537822 < x < 119.57592943098413

    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-cbrt0.0

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

    4. Applied associate-/l*0.0

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

    6. Applied frac-sub0.0

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

    7. Applied frac-add0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -136.1881966537822 \lor \neg \left(x \le 119.57592943098413\right):\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, 2 \cdot {x}^{\left(-3\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, \frac{x - 1}{\sqrt[3]{1}}, \left(\left(x + 1\right) \cdot x\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \frac{x - 1}{\sqrt[3]{1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))