Average Error: 9.6 → 0.5
Time: 3.2s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -218778.51422173358:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \frac{\frac{\frac{2}{x \cdot x}}{x - 1}}{\frac{1}{x + 1} - \frac{2}{x}}\\ \mathbf{elif}\;x \leq 299129.8760379891:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \frac{\frac{2}{x}}{x}}{\left(x - 1\right) \cdot \left(\frac{1}{x + 1} + \frac{2}{x}\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -218778.51422173358:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \frac{\frac{\frac{2}{x \cdot x}}{x - 1}}{\frac{1}{x + 1} - \frac{2}{x}}\\

\mathbf{elif}\;x \leq 299129.8760379891:\\
\;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \frac{\frac{2}{x}}{x}}{\left(x - 1\right) \cdot \left(\frac{1}{x + 1} + \frac{2}{x}\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 -218778.51422173358)
   (*
    (- (/ 1.0 (+ x 1.0)) (/ 2.0 x))
    (/ (/ (/ 2.0 (* x x)) (- x 1.0)) (- (/ 1.0 (+ x 1.0)) (/ 2.0 x))))
   (if (<= x 299129.8760379891)
     (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))
     (/
      (* (+ (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ (/ 2.0 x) x))
      (* (- x 1.0) (+ (/ 1.0 (+ x 1.0)) (/ 2.0 x)))))))
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 <= -218778.51422173358)) {
		tmp = ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) * (((2.0 / ((double) (x * x))) / ((double) (x - 1.0))) / ((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))))));
	} else {
		double tmp_1;
		if ((x <= 299129.8760379891)) {
			tmp_1 = ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + (1.0 / ((double) (x - 1.0)))));
		} else {
			tmp_1 = (((double) (((double) ((1.0 / ((double) (x + 1.0))) + (2.0 / x))) * ((2.0 / x) / x))) / ((double) (((double) (x - 1.0)) * ((double) ((1.0 / ((double) (x + 1.0))) + (2.0 / x))))));
		}
		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

Original9.6
Target0.2
Herbie0.5
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -218778.514221733582

    1. Initial program 19.8

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

    3. Applied flip--_binary6453.7

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

    4. Applied frac-add_binary6454.5

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

    5. Simplified25.4

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

    6. Simplified25.4

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

    7. Taylor expanded around -inf 0.7

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

    8. Simplified0.7

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

    10. Applied flip-+_binary6432.1

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

    11. Applied associate-*r/_binary6432.1

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

    12. Applied associate-/r/_binary6432.0

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

    13. Simplified0.7

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

    if -218778.514221733582 < x < 299129.876037989103

    1. Initial program 0.3

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

    if 299129.876037989103 < x

    1. Initial program 19.4

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

    3. Applied flip--_binary6454.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

    4. Applied frac-add_binary6454.9

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

    5. Simplified24.9

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

    6. Simplified24.9

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

    7. Taylor expanded around -inf 0.8

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

    8. Simplified0.8

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

    10. Applied associate-/r*_binary640.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -218778.51422173358:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) \cdot \frac{\frac{\frac{2}{x \cdot x}}{x - 1}}{\frac{1}{x + 1} - \frac{2}{x}}\\ \mathbf{elif}\;x \leq 299129.8760379891:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \frac{\frac{2}{x}}{x}}{\left(x - 1\right) \cdot \left(\frac{1}{x + 1} + \frac{2}{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))