Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 8.6s

Precision: binary64

Cost: 2624

Math

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

↓

\[\begin{array}{l} t_1 := 8 + \frac{-4}{1 + t}\\ 1 + \frac{-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}}{36 - \frac{\frac{t_1}{1 + t}}{\frac{1 + t}{t_1}}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (-
  1.0
  (/
   1.0
   (+
    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 (+ 8.0 (/ -4.0 (+ 1.0 t)))))
   (+
    1.0
    (/
     (+ -6.0 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)))
     (- 36.0 (/ (/ t_1 (+ 1.0 t)) (/ (+ 1.0 t) t_1)))))))

C

double code(double t) {
	return 1.0 - (1.0 / (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 = 8.0 + (-4.0 / (1.0 + t));
	return 1.0 + ((-6.0 + (((4.0 / (1.0 + t)) + -8.0) / (1.0 + t))) / (36.0 - ((t_1 / (1.0 + t)) / ((1.0 + t) / t_1))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 - (1.0d0 / (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 = 8.0d0 + ((-4.0d0) / (1.0d0 + t))
    code = 1.0d0 + (((-6.0d0) + (((4.0d0 / (1.0d0 + t)) + (-8.0d0)) / (1.0d0 + t))) / (36.0d0 - ((t_1 / (1.0d0 + t)) / ((1.0d0 + t) / t_1))))
end function

Java

public static double code(double t) {
	return 1.0 - (1.0 / (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 = 8.0 + (-4.0 / (1.0 + t));
	return 1.0 + ((-6.0 + (((4.0 / (1.0 + t)) + -8.0) / (1.0 + t))) / (36.0 - ((t_1 / (1.0 + t)) / ((1.0 + t) / t_1))));
}

Python

def code(t):
	return 1.0 - (1.0 / (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 = 8.0 + (-4.0 / (1.0 + t))
	return 1.0 + ((-6.0 + (((4.0 / (1.0 + t)) + -8.0) / (1.0 + t))) / (36.0 - ((t_1 / (1.0 + t)) / ((1.0 + t) / t_1))))

Julia

function code(t)
	return Float64(1.0 - Float64(1.0 / 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(8.0 + Float64(-4.0 / Float64(1.0 + t)))
	return Float64(1.0 + Float64(Float64(-6.0 + Float64(Float64(Float64(4.0 / Float64(1.0 + t)) + -8.0) / Float64(1.0 + t))) / Float64(36.0 - Float64(Float64(t_1 / Float64(1.0 + t)) / Float64(Float64(1.0 + t) / t_1)))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 - (1.0 / (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 = 8.0 + (-4.0 / (1.0 + t));
	tmp = 1.0 + ((-6.0 + (((4.0 / (1.0 + t)) + -8.0) / (1.0 + t))) / (36.0 - ((t_1 / (1.0 + t)) / ((1.0 + t) / t_1))));
end

Wolfram

code[t_] := N[(1.0 - N[(1.0 / 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]), $MachinePrecision]

↓

code[t_] := Block[{t$95$1 = N[(8.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[(-6.0 + N[(N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + -8.0), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(36.0 - N[(N[(t$95$1 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

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

3. Applied egg-rr 0.0

   \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{-6 - \frac{\frac{4}{1 + t} + -8}{1 + t}}\right)\right)} \]

4. Applied egg-rr 0.0

   \[\leadsto 1 + \color{blue}{\frac{1}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \cdot \left(-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}\right)} \]

5. Simplified 0.0

   \[\leadsto 1 + \color{blue}{\frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{8 + \frac{-4}{t + 1}}{t + 1}}} \]
   Proof

   [Start] 0.0	\[ 1 + \frac{1}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \cdot \left(-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}\right) \]
   associate-*l/ [=>] 0.0	\[ 1 + \color{blue}{\frac{1 \cdot \left(-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}\right)}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}}} \]
   *-lft-identity [=>] 0.0	\[ 1 + \frac{\color{blue}{-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}}}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\color{blue}{\frac{\frac{4}{1 + t} + -8}{1 + t} + -6}}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{\color{blue}{t + 1}} + -8}{1 + t} + -6}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{\color{blue}{t + 1}} + -6}{36 - \frac{8 - \frac{4}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   sub-neg [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{\color{blue}{8 + \left(-\frac{4}{1 + t}\right)}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   distribute-neg-frac [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \color{blue}{\frac{-4}{1 + t}}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{\color{blue}{-4}}{1 + t}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{\color{blue}{t + 1}}}{1 + t} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{\color{blue}{t + 1}} \cdot \frac{8 - \frac{4}{1 + t}}{1 + t}} \]
   sub-neg [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{\color{blue}{8 + \left(-\frac{4}{1 + t}\right)}}{1 + t}} \]
   distribute-neg-frac [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{8 + \color{blue}{\frac{-4}{1 + t}}}{1 + t}} \]
   metadata-eval [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{8 + \frac{\color{blue}{-4}}{1 + t}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{8 + \frac{-4}{\color{blue}{t + 1}}}{1 + t}} \]
   +-commutative [=>] 0.0	\[ 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \frac{8 + \frac{-4}{t + 1}}{t + 1} \cdot \frac{8 + \frac{-4}{t + 1}}{\color{blue}{t + 1}}} \]

6. Applied egg-rr 0.0

   \[\leadsto 1 + \frac{\frac{\frac{4}{t + 1} + -8}{t + 1} + -6}{36 - \color{blue}{\frac{\frac{8 + \frac{-4}{t + 1}}{t + 1}}{\frac{t + 1}{8 + \frac{-4}{t + 1}}}}} \]

7. Final simplification 0.0

   \[\leadsto 1 + \frac{-6 + \frac{\frac{4}{1 + t} + -8}{1 + t}}{36 - \frac{\frac{8 + \frac{-4}{1 + t}}{1 + t}}{\frac{1 + t}{8 + \frac{-4}{1 + t}}}} \]

Alternatives

Alternative 1
Error	0.0
Cost	2624

\[\begin{array}{l} t_1 := \frac{\frac{4}{1 + t} + -8}{1 + t}\\ 1 + \frac{-6 + t_1}{36 + \frac{8 + \frac{-4}{1 + t}}{1 + t} \cdot t_1} \end{array} \]

Alternative 2
Error	0.0
Cost	1088

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

Alternative 3
Error	0.4
Cost	969

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

Alternative 4
Error	0.4
Cost	968

\[\begin{array}{l} t_1 := \frac{-0.037037037037037035}{t \cdot t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;1 + \left(\frac{-0.2222222222222222}{t} - \left(0.16666666666666666 + t_1\right)\right)\\ \mathbf{elif}\;t \leq 0.24:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \left(\frac{-0.2222222222222222}{t} - t_1\right)\\ \end{array} \]

Alternative 5
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 6
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 7
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 8
Error	26.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022364 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))