Average Error: 9.3 → 0.2
Time: 3.2s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -87.53679547805154:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 94.7577384407541:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -87.53679547805154:\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\

\mathbf{elif}\;x \leq 94.7577384407541:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\

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

\end{array}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -87.53679547805154)
   (+ (/ 2.0 (pow x 7.0)) (+ (/ 2.0 (pow x 5.0)) (/ (/ 2.0 (* x x)) x)))
   (if (<= x 94.7577384407541)
     (+
      (- (/ 1.0 (+ x 1.0)) (/ 2.0 x))
      (/ (/ 1.0 (* (cbrt (- x 1.0)) (cbrt (- x 1.0)))) (cbrt (- x 1.0))))
     (+ (/ 2.0 (pow x 7.0)) (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 3.0)))))))
double code(double x) {
	return ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + (1.0 / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -87.53679547805154)) {
		tmp = ((double) ((2.0 / ((double) pow(x, 7.0))) + ((double) ((2.0 / ((double) pow(x, 5.0))) + ((2.0 / ((double) (x * x))) / x)))));
	} else {
		double tmp_1;
		if ((x <= 94.7577384407541)) {
			tmp_1 = ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + ((1.0 / ((double) (((double) cbrt(((double) (x - 1.0)))) * ((double) cbrt(((double) (x - 1.0))))))) / ((double) cbrt(((double) (x - 1.0)))))));
		} else {
			tmp_1 = ((double) ((2.0 / ((double) pow(x, 7.0))) + ((double) ((2.0 / ((double) pow(x, 5.0))) + (2.0 / ((double) pow(x, 3.0)))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -87.536795478051545

    1. Initial program 18.8

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    5. Applied unpow3_binary640.7

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

    6. Applied associate-/r*_binary640.1

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

    if -87.536795478051545 < x < 94.7577384407540961

    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-cbrt_binary640.1

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

    4. Applied associate-/r*_binary640.1

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

    if 94.7577384407540961 < x

    1. Initial program 19.0

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -87.53679547805154:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 94.7577384407541:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))