Average Error: 9.5 → 10.0
Time: 11.8s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3270199621315663 \cdot 10^{+154} \lor \neg \left(x \leq 1.3274650429408704 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \left(x + 1\right) \cdot \frac{\frac{1}{x - 1}}{x \cdot x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{x \cdot x - 1}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.3270199621315663 \cdot 10^{+154} \lor \neg \left(x \leq 1.3274650429408704 \cdot 10^{+154}\right):\\
\;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \left(x + 1\right) \cdot \frac{\frac{1}{x - 1}}{x \cdot x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{x \cdot 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 -1.3270199621315663e+154) (not (<= x 1.3274650429408704e+154)))
   (/
    (-
     (* (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)))
     (* (+ x 1.0) (/ (/ 1.0 (- x 1.0)) (- (* x x) 1.0))))
    (- (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
   (*
    (/ 1.0 x)
    (/
     (+ (* (- x 1.0) (- x (* (+ x 1.0) 2.0))) (* x (+ x 1.0)))
     (- (* x 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 <= -1.3270199621315663e+154) || !(x <= 1.3274650429408704e+154)) {
		tmp = ((((1.0 / (x + 1.0)) - (2.0 / x)) * ((1.0 / (x + 1.0)) - (2.0 / x))) - ((x + 1.0) * ((1.0 / (x - 1.0)) / ((x * x) - 1.0)))) / (((1.0 / (x + 1.0)) - (2.0 / x)) - (1.0 / (x - 1.0)));
	} else {
		tmp = (1.0 / x) * ((((x - 1.0) * (x - ((x + 1.0) * 2.0))) + (x * (x + 1.0))) / ((x * 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

Original9.5
Target0.3
Herbie10.0
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3270199621315663e154 or 1.32746504294087043e154 < x

    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-+_binary640

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

    4. Simplified0

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

    5. Simplified0

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

    7. Applied flip--_binary642.8

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

    8. Applied associate-/r/_binary642.8

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

    9. Simplified2.8

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

    if -1.3270199621315663e154 < x < 1.32746504294087043e154

    1. Initial program 12.8

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

    3. Applied frac-sub_binary6414.7

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

    4. Applied frac-add_binary6412.4

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

    5. Simplified12.4

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

    6. Simplified12.4

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

    8. Applied *-un-lft-identity_binary6412.4

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

    9. Applied distribute-lft-neg-in_binary6412.4

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

    10. Applied unpow3_binary6412.4

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

    11. Applied distribute-rgt-out_binary6412.4

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

    12. Applied *-un-lft-identity_binary6412.4

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

    13. Applied times-frac_binary6412.4

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

    14. Simplified12.4

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3270199621315663 \cdot 10^{+154} \lor \neg \left(x \leq 1.3274650429408704 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \left(\frac{1}{x + 1} - \frac{2}{x}\right) - \left(x + 1\right) \cdot \frac{\frac{1}{x - 1}}{x \cdot x - 1}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{x \cdot x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))