Average Error: 10.0 → 0.6
Time: 5.7s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -4353305.628193324:\\ \;\;\;\;\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 0:\\ \;\;\;\;-2 \cdot \left(-{x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -4353305.628193324:\\
\;\;\;\;\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\\

\mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 0:\\
\;\;\;\;-2 \cdot \left(-{x}^{-3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(1 + x\right)\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 (<=
      (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))
      -4353305.628193324)
   (- (/ 1.0 (+ 1.0 x)) (- (/ 2.0 x) (/ 1.0 (- x 1.0))))
   (if (<= (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (- x 1.0))) 0.0)
     (* -2.0 (- (pow x -3.0)))
     (/
      (+ (* (- x 1.0) (- x (* (+ 1.0 x) 2.0))) (* x (+ 1.0 x)))
      (* (- x 1.0) (* x (+ 1.0 x)))))))
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 ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= -4353305.628193324) {
		tmp = (1.0 / (1.0 + x)) - ((2.0 / x) - (1.0 / (x - 1.0)));
	} else if ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= 0.0) {
		tmp = -2.0 * -pow(x, -3.0);
	} else {
		tmp = (((x - 1.0) * (x - ((1.0 + x) * 2.0))) + (x * (1.0 + x))) / ((x - 1.0) * (x * (1.0 + x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.6
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4353305.6281933244

    1. Initial program 0.0

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

    3. Applied associate-+l-_binary64_10360.0

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

    4. Simplified0.0

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

    if -4353305.6281933244 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

    1. Initial program 19.3

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

    3. Applied add-exp-log_binary64_113920.4

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

    4. Simplified20.4

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified0.8

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

    if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

    1. Initial program 1.4

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

    3. Applied frac-sub_binary64_11101.5

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

    4. Applied frac-add_binary64_11090.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -4353305.628193324:\\ \;\;\;\;\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 0:\\ \;\;\;\;-2 \cdot \left(-{x}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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))))