Average Error: 10.0 → 0.5
Time: 8.0s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -7.643396633563098:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{-1}{1 - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 4.5263323118901674 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \left(\frac{1}{x} \cdot -2 + {x}^{3} \cdot -2\right)\\ \end{array}\]
\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 -7.643396633563098:\\
\;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{-1}{1 - x}\\

\mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 4.5263323118901674 \cdot 10^{-21}:\\
\;\;\;\;\left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot -2 + \left(\frac{1}{x} \cdot -2 + {x}^{3} \cdot -2\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)))
      -7.643396633563098)
   (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ -1.0 (- 1.0 x)))
   (if (<=
        (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))
        4.5263323118901674e-21)
     (+
      (+ (+ (/ (/ 2.0 x) (* x x)) (/ 2.0 (pow x 5.0))) (/ 2.0 (pow x 7.0)))
      (/ 2.0 (pow x 9.0)))
     (+ (* x -2.0) (+ (* (/ 1.0 x) -2.0) (* (pow x 3.0) -2.0))))))
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 ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= -7.643396633563098) {
		tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (-1.0 / (1.0 - x));
	} else if ((((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x - 1.0))) <= 4.5263323118901674e-21) {
		tmp = ((((2.0 / x) / (x * x)) + (2.0 / pow(x, 5.0))) + (2.0 / pow(x, 7.0))) + (2.0 / pow(x, 9.0));
	} else {
		tmp = (x * -2.0) + (((1.0 / x) * -2.0) + (pow(x, 3.0) * -2.0));
	}
	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
Herbie0.5
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -7.6433966335630981

    1. Initial program 0.0

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

    3. Applied frac-2neg_binary64_11120.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -7.6433966335630981 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.52633e-21

    1. Initial program 19.9

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{9}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    3. Simplified0.6

      \[\leadsto \color{blue}{\left(\left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}}\]
    4. Using strategy rm

    5. Applied cube-mult_binary64_11310.6

      \[\leadsto \left(\left(\frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}\]

    6. Applied associate-/r*_binary64_10450.2

      \[\leadsto \left(\left(\color{blue}{\frac{\frac{2}{x}}{x \cdot x}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}\]

    if 4.52633e-21 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

    1. Initial program 0.5

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

    2. Taylor expanded around 0 1.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -7.643396633563098:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{-1}{1 - x}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 4.5263323118901674 \cdot 10^{-21}:\\ \;\;\;\;\left(\left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{7}}\right) + \frac{2}{{x}^{9}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \left(\frac{1}{x} \cdot -2 + {x}^{3} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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))))