Average Error: 9.9 → 0.2
Time: 5.2s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -108.93030297428587:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 109.900664385492959:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -108.93030297428587:\\
\;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\

\mathbf{elif}\;x \le 109.900664385492959:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\

\end{array}
double code(double x) {
	return ((double) (((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (2.0 / x)))) + ((double) (1.0 / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -108.93030297428587)) {
		VAR = ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 7.0)))), ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 5.0)))), ((double) (((double) (2.0 / x)) / ((double) (x * x))))))));
	} else {
		double VAR_1;
		if ((x <= 109.90066438549296)) {
			VAR_1 = ((double) (((double) fma(((double) (1.0 / ((double) (((double) (x * x)) - ((double) (1.0 * 1.0)))))), ((double) (x - 1.0)), ((double) -(((double) (2.0 / x)))))) + ((double) (1.0 / ((double) (x - 1.0))))));
		} else {
			VAR_1 = ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 7.0)))), ((double) fma(2.0, ((double) (1.0 / ((double) pow(x, 5.0)))), ((double) (2.0 / ((double) pow(x, 3.0))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -108.93030297428587

    1. Initial program 20.7

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

    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

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

    5. Applied cube-mult0.5

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

    6. Applied associate-/r*0.1

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

    if -108.93030297428587 < x < 109.90066438549296

    1. Initial program 0.0

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

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Applied fma-neg0.0

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

    if 109.90066438549296 < x

    1. Initial program 19.3

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -108.93030297428587:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{\frac{2}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 109.900664385492959:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x - 1, -\frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{{x}^{7}}, \mathsf{fma}\left(2, \frac{1}{{x}^{5}}, \frac{2}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))))