Kahan p13 Example 2

Average Error: 0.0 → 0.0

Time: 8.7s

Precision: binary64

Cost: 1728

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ \frac{t_1 + 5}{t_1 + 6} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   2.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))

↓

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t))))
   (/ (+ t_1 5.0) (+ t_1 6.0))))

C

double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}

↓

double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (t_1 + 5.0) / (t_1 + 6.0);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (1.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))))) / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t))))))
end function

↓

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t)
    code = (t_1 + 5.0d0) / (t_1 + 6.0d0)
end function

Java

public static double code(double t) {
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}

↓

public static double code(double t) {
	double t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	return (t_1 + 5.0) / (t_1 + 6.0);
}

Python

def code(t):
	return (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))))

↓

def code(t):
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)
	return (t_1 + 5.0) / (t_1 + 6.0)

Julia

function code(t)
	return Float64(Float64(1.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))) / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))))))
end

↓

function code(t)
	t_1 = Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t))
	return Float64(Float64(t_1 + 5.0) / Float64(t_1 + 6.0))
end

MATLAB

function tmp = code(t)
	tmp = (1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
end

↓

function tmp = code(t)
	t_1 = ((4.0 / (1.0 + t)) + -8.0) / (1.0 + t);
	tmp = (t_1 + 5.0) / (t_1 + 6.0);
end

Wolfram

code[t_] := N[(N[(1.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := Block[{t$95$1 = N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 + 5.0), $MachinePrecision] / N[(t$95$1 + 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}} \]
   Proof

   [Start] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.0	\[ \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-rgt-in [=>] 0.0	\[ \frac{1 + \color{blue}{\left(2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+r+ [=>] 0.0	\[ \frac{\color{blue}{\left(1 + 2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.0	\[ \frac{\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [=>] 0.0	\[ \frac{\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + \color{blue}{\left(2 \cdot 2 + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+r+ [=>] 0.0	\[ \frac{\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{\left(\left(1 + 2 \cdot 2\right) + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.0	\[ \frac{\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{\left(2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot 2\right)\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+r+ [=>] 0.0	\[ \frac{\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(1 + 2 \cdot 2\right)}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-rgt-neg-out [=>] 0.0	\[ \frac{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{\left(-2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   unsub-neg [=>] 0.0	\[ \frac{\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) - 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   cancel-sign-sub-inv [=>] 0.0	\[ \frac{\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-commutative [=>] 0.0	\[ \frac{\left(\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   neg-mul-1 [=>] 0.0	\[ \frac{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-*r* [=>] 0.0	\[ \frac{\left(\color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} + \left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-rgt-out [=>] 0.0	\[ \frac{\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1 + \left(-2\right)\right)} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [<=] 0.0	\[ \frac{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \color{blue}{\left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right)} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-/l/ [=>] 0.3	\[ \frac{\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}} \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-*l/ [=>] 0.3	\[ \frac{\color{blue}{\frac{2 \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right)}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-rgt-in [=>] 0.3	\[ \frac{\frac{\color{blue}{\left(-2\right) \cdot 2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) \cdot 2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-*l* [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{-2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-commutative [<=] 0.3	\[ \frac{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(-2\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [<=] 0.3	\[ \frac{\frac{\color{blue}{\left(-2\right) \cdot \left(2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \left(\color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \left(\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2\right)} + 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+l+ [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 + 2\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{4}\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{\frac{\left(-2\right) \cdot \left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{2 \cdot 2}\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [=>] 0.3	\[ \frac{\frac{\color{blue}{\left(-2\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-neg-in [<=] 0.3	\[ \frac{\frac{\color{blue}{\left(-2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-rgt-neg-out [=>] 0.3	\[ \frac{\frac{\left(-\color{blue}{\left(-2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   remove-double-neg [=>] 0.3	\[ \frac{\frac{\color{blue}{2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-/l/ [=>] 0.2	\[ \frac{\frac{2 \cdot \color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-*r/ [=>] 0.2	\[ \frac{\frac{\color{blue}{\frac{2 \cdot 2}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-commutative [=>] 0.2	\[ \frac{\frac{\frac{2 \cdot 2}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{\frac{\color{blue}{4}}{t \cdot \left(1 + \frac{1}{t}\right)} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.2	\[ \frac{\frac{\frac{4}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [=>] 0.2	\[ \frac{\frac{\frac{4}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   rgt-mult-inverse [=>] 0.3	\[ \frac{\frac{\frac{4}{\color{blue}{1} + t \cdot 1} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-rgt-identity [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + \color{blue}{t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + \color{blue}{-2} \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + -2 \cdot \color{blue}{4}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + \color{blue}{-8}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-commutative [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + -8}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [=>] 0.3	\[ \frac{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   rgt-mult-inverse [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{\color{blue}{1} + t \cdot 1} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   *-rgt-identity [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + \color{blue}{t}} + \left(1 + 2 \cdot 2\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + \left(1 + \color{blue}{4}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + \color{blue}{5}}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   sub-neg [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}} \]
   distribute-rgt-in [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{2 + \color{blue}{\left(2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}} \]
   associate-+r+ [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\left(2 + 2 \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}} \]
   sub-neg [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(2 + 2 \cdot \color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   distribute-lft-in [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(2 + \color{blue}{\left(2 \cdot 2 + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+r+ [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\left(\left(2 + 2 \cdot 2\right) + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   associate-+l+ [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\left(2 + 2 \cdot 2\right) + \left(2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}} \]
   +-commutative [<=] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(2 + 2 \cdot 2\right) + \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}} \]
   +-commutative [<=] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(2 + 2 \cdot 2\right)}} \]
   +-commutative [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\left(2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + \left(2 + 2 \cdot 2\right)} \]
   distribute-rgt-neg-out [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(\color{blue}{\left(-2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-neg-in [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(\color{blue}{\left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) + \left(2 + 2 \cdot 2\right)} \]
   *-commutative [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(\left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + \color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + \left(2 + 2 \cdot 2\right)} \]
   neg-mul-1 [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(\left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + \left(2 + 2 \cdot 2\right)} \]
   associate-*r* [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\left(\left(-2\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + \color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}}\right) + \left(2 + 2 \cdot 2\right)} \]
   distribute-rgt-out [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right)} + \left(2 + 2 \cdot 2\right)} \]
   associate-/l/ [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}} \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) + \left(2 + 2 \cdot 2\right)} \]
   associate-*l/ [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\color{blue}{\frac{2 \cdot \left(\left(-2\right) + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right)}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(2 + 2 \cdot 2\right)} \]
   distribute-rgt-in [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{\left(-2\right) \cdot 2 + \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot -1\right) \cdot 2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   associate-*l* [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(-1 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{-2}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [<=] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot 2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(-2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   *-commutative [<=] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot 2 + \color{blue}{\left(-2\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-in [<=] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{\left(-2\right) \cdot \left(2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   +-commutative [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   sub-neg [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \left(\color{blue}{\left(2 + \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   +-commutative [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \left(\color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2\right)} + 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   associate-+l+ [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \color{blue}{\left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(2 + 2\right)\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{4}\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [<=] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-2\right) \cdot \left(\left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \color{blue}{2 \cdot 2}\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-in [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{\left(-2\right) \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-neg-in [<=] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{\left(-2 \cdot \left(-\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   distribute-rgt-neg-out [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\left(-\color{blue}{\left(-2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\right) + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   remove-double-neg [=>] 0.2	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   associate-/l/ [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{2 \cdot \color{blue}{\frac{2}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   associate-*r/ [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\color{blue}{\frac{2 \cdot 2}{\left(1 + \frac{1}{t}\right) \cdot t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   *-commutative [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{2 \cdot 2}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{\color{blue}{4}}{t \cdot \left(1 + \frac{1}{t}\right)} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-in [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   rgt-mult-inverse [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{\color{blue}{1} + t \cdot 1} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   *-rgt-identity [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + \color{blue}{t}} + \left(-2\right) \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + \color{blue}{-2} \cdot \left(2 \cdot 2\right)}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -2 \cdot \color{blue}{4}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + \color{blue}{-8}}{\left(1 + \frac{1}{t}\right) \cdot t} + \left(2 + 2 \cdot 2\right)} \]
   *-commutative [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \left(1 + \frac{1}{t}\right)}} + \left(2 + 2 \cdot 2\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{t \cdot \color{blue}{\left(\frac{1}{t} + 1\right)}} + \left(2 + 2 \cdot 2\right)} \]
   distribute-lft-in [=>] 0.1	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{t \cdot \frac{1}{t} + t \cdot 1}} + \left(2 + 2 \cdot 2\right)} \]
   rgt-mult-inverse [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{\color{blue}{1} + t \cdot 1} + \left(2 + 2 \cdot 2\right)} \]
   *-rgt-identity [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + \color{blue}{t}} + \left(2 + 2 \cdot 2\right)} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + \left(2 + \color{blue}{4}\right)} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + \color{blue}{6}} \]

3. Final simplification 0.0

   \[\leadsto \frac{\frac{\frac{4}{1 + t} + -8}{1 + t} + 5}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6} \]

Alternatives

Alternative 1
Error	0.4
Cost	969

\[\begin{array}{l} \mathbf{if}\;t \leq -0.82 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;0.8333333333333334 + \left(\frac{0.037037037037037035}{t \cdot t} + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 2
Error	0.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -0.8 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;t \cdot t + 0.5\\ \end{array} \]

Alternative 3
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.56:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 4
Error	1.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \]

Alternative 5
Error	26.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022364 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))