Average Error: 10.1 → 0.1
Time: 2.8s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -106.76785359664669 \lor \neg \left(x \leq 126.59925884127742\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -106.76785359664669 \lor \neg \left(x \leq 126.59925884127742\right):\\
\;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\

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

\end{array}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -106.76785359664669) (not (<= x 126.59925884127742)))
   (+ (/ 2.0 (pow x 7.0)) (+ (/ 2.0 (pow x 5.0)) (/ (/ (/ 2.0 x) x) x)))
   (+
    (- (/ 1.0 (+ x 1.0)) (/ 2.0 x))
    (*
     (/ 1.0 (* (cbrt (- x 1.0)) (cbrt (- x 1.0))))
     (/ 1.0 (cbrt (- x 1.0)))))))
double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -106.76785359664669) || !(x <= 126.59925884127742)) {
		tmp = (2.0 / pow(x, 7.0)) + ((2.0 / pow(x, 5.0)) + (((2.0 / x) / x) / x));
	} else {
		tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + ((1.0 / (cbrt(x - 1.0) * cbrt(x - 1.0))) * (1.0 / cbrt(x - 1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.1
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -106.76785359664669 or 126.59925884127742 < x

    1. Initial program 20.1

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

    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

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

    5. Applied unpow3_binary640.6

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

    6. Applied associate-/r*_binary640.1

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

    8. Applied associate-/r*_binary640.1

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

    if -106.76785359664669 < x < 126.59925884127742

    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_binary640.0

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

    4. Applied add-sqr-sqrt_binary640.0

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

    5. Applied times-frac_binary640.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -106.76785359664669 \lor \neg \left(x \leq 126.59925884127742\right):\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))))