Kahan p13 Example 3

Percentage Accurate: 99.9% → 100.0%

Time: 7.9s

Alternatives: 11

Speedup: 0.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ 1 - \frac{1}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return 1.0 - (1.0 / (2.0 + (t_1 * t_1)))

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + \frac{-2}{1 + t}\\ 1 + \frac{-1}{\mathsf{fma}\left(t\_1, t\_1, 2\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (/ -2.0 (+ 1.0 t))))) (+ 1.0 (/ -1.0 (fma t_1 t_1 2.0)))))

C

double code(double t) {
	double t_1 = 2.0 + (-2.0 / (1.0 + t));
	return 1.0 + (-1.0 / fma(t_1, t_1, 2.0));
}

Julia

function code(t)
	t_1 = Float64(2.0 + Float64(-2.0 / Float64(1.0 + t)))
	return Float64(1.0 + Float64(-1.0 / fma(t_1, t_1, 2.0)))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 + N[(-2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(t$95$1 * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{\mathsf{fma}\left(2 + \frac{-2}{1 + t}, 2 + \frac{-2}{1 + t}, 2\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{2 + \left(2 + \frac{\frac{2}{t}}{-1 - \frac{1}{t}}\right) \cdot \left(2 + \frac{1}{\left(-1 - t\right) \cdot 0.5}\right)} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+
  1.0
  (/
   -1.0
   (+
    2.0
    (*
     (+ 2.0 (/ (/ 2.0 t) (- -1.0 (/ 1.0 t))))
     (+ 2.0 (/ 1.0 (* (- -1.0 t) 0.5))))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 - (1.0 / t)))) * (2.0 + (1.0 / ((-1.0 - t) * 0.5))))));
}

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 + (1.0d0 / (((-1.0d0) - t) * 0.5d0))))))
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 + (1.0 / ((-1.0 - t) * 0.5))))));
}

Python

def code(t):
	return 1.0 + (-1.0 / (2.0 + ((2.0 + ((2.0 / t) / (-1.0 - (1.0 / t)))) * (2.0 + (1.0 / ((-1.0 - t) * 0.5))))))

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(1.0 / Float64(Float64(-1.0 - t) * 0.5)))))))
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 + (1.0 / ((-1.0 - t) * 0.5))))));
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[(1.0 / N[(N[(-1.0 - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. clear-num 100.0%

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

   5. div-inv 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
5. Step-by-step derivation

   1. unpow-1 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

   2. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

   3. *-commutative 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

   4. associate-*l* 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

   5. distribute-lft-in 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

   6. *-rgt-identity 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

   7. rgt-mult-inverse 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

6. Simplified 100.0%

   \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

7. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{2 + \left(2 + \frac{\frac{2}{t}}{-1 - \frac{1}{t}}\right) \cdot \left(2 + \frac{1}{\left(-1 - t\right) \cdot 0.5}\right)} \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{4}{1 + t} - 4\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -1.2)
   (-
    1.0
    (+
     (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)
     0.16666666666666666))
   (if (<= t 1.55)
     (+ 1.0 (/ 1.0 (- (* t (- (/ 4.0 (+ 1.0 t)) 4.0)) 2.0)))
     (+ 1.0 (/ -1.0 (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double tmp;
	if (t <= -1.2) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.55) {
		tmp = 1.0 + (1.0 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.2d0)) then
        tmp = 1.0d0 - (((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t) + 0.16666666666666666d0)
    else if (t <= 1.55d0) then
        tmp = 1.0d0 + (1.0d0 / ((t * ((4.0d0 / (1.0d0 + t)) - 4.0d0)) - 2.0d0))
    else
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -1.2) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.55) {
		tmp = 1.0 + (1.0 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -1.2:
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666)
	elif t <= 1.55:
		tmp = 1.0 + (1.0 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0))
	else:
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -1.2)
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t) + 0.16666666666666666));
	elseif (t <= 1.55)
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(t * Float64(Float64(4.0 / Float64(1.0 + t)) - 4.0)) - 2.0)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -1.2)
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	elseif (t <= 1.55)
		tmp = 1.0 + (1.0 / ((t * ((4.0 / (1.0 + t)) - 4.0)) - 2.0));
	else
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -1.2], N[(1.0 - N[(N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55], N[(1.0 + N[(1.0 / N[(N[(t * N[(N[(4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - \frac{0.037037037037037035}{{t}^{2}}\right)} \]
   4. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto 1 - \color{blue}{\left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) + 0.16666666666666666\right)} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right) + 0.16666666666666666\right) \]

      4. associate-/r* 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right) + 0.16666666666666666\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t}\right) + 0.16666666666666666\right) \]

      6. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t}\right) + 0.16666666666666666\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      8. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      9. div-sub 100.0%

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} + 0.16666666666666666\right) \]

      10. sub-neg 100.0%

          \[\leadsto 1 - \left(\frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} + 0.16666666666666666\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} + 0.16666666666666666\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} + 0.16666666666666666\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} + 0.16666666666666666\right) \]

      14. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} + 0.16666666666666666\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)} \]

   if -1.19999999999999996 < t < 1.55000000000000004

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

      4. associate-*l* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

      5. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

      6. *-rgt-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

      7. rgt-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(t \cdot 2\right)}} \]
   10. Step-by-step derivation

       1. clear-num 100.0%

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

       2. inv-pow 100.0%

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

       3. div-inv 100.0%

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

       4. clear-num 100.0%

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

       5. div-inv 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(t \cdot 2\right)} \]
   12. Step-by-step derivation

       1. unpow-1 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

       2. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

       3. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

       4. associate-*l* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

       5. distribute-lft-in 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

       6. *-rgt-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

       7. rgt-mult-inverse 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   13. Simplified 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(t \cdot 2\right)} \]
   14. Step-by-step derivation

       1. add-log-exp 100.0%

          \[\leadsto 1 - \frac{1}{2 + \color{blue}{\log \left(e^{\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot \left(t \cdot 2\right)}\right)}} \]

       2. *-un-lft-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \log \color{blue}{\left(1 \cdot e^{\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot \left(t \cdot 2\right)}\right)}} \]

       3. log-prod 100.0%

          \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(\log 1 + \log \left(e^{\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot \left(t \cdot 2\right)}\right)\right)}} \]

       4. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{0} + \log \left(e^{\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot \left(t \cdot 2\right)}\right)\right)} \]

       5. add-log-exp 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(0 + \color{blue}{\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot \left(t \cdot 2\right)}\right)} \]

       6. associate-*r* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(0 + \color{blue}{\left(\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot t\right) \cdot 2}\right)} \]

       7. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(0 + \color{blue}{2 \cdot \left(\left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right) \cdot t\right)}\right)} \]

       8. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \color{blue}{\left(t \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + 1\right)}\right)\right)}\right)} \]

       9. sub-neg 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \color{blue}{\left(2 + \left(-\frac{1}{0.5 \cdot \left(t + 1\right)}\right)\right)}\right)\right)} \]

       10. associate-/r* 100.0%

           \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \left(2 + \left(-\color{blue}{\frac{\frac{1}{0.5}}{t + 1}}\right)\right)\right)\right)} \]

       11. metadata-eval 100.0%

           \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \left(2 + \left(-\frac{\color{blue}{2}}{t + 1}\right)\right)\right)\right)} \]

       12. distribute-neg-frac 100.0%

           \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \left(2 + \color{blue}{\frac{-2}{t + 1}}\right)\right)\right)} \]

       13. metadata-eval 100.0%

           \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \left(2 + \frac{\color{blue}{-2}}{t + 1}\right)\right)\right)} \]

       14. +-commutative 100.0%

           \[\leadsto 1 - \frac{1}{2 + \left(0 + 2 \cdot \left(t \cdot \left(2 + \frac{-2}{\color{blue}{1 + t}}\right)\right)\right)} \]

   15. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(0 + 2 \cdot \left(t \cdot \left(2 + \frac{-2}{1 + t}\right)\right)\right)}} \]
   16. Step-by-step derivation

       1. +-lft-identity 100.0%

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

       2. associate-*r* 100.0%

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

       3. distribute-rgt-out 100.0%

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

       4. associate-*r* 100.0%

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

       5. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(\color{blue}{4} \cdot t + \frac{-2}{1 + t} \cdot \left(2 \cdot t\right)\right)} \]

       6. associate-*r* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(4 \cdot t + \color{blue}{\left(\frac{-2}{1 + t} \cdot 2\right) \cdot t}\right)} \]

       7. distribute-rgt-out 100.0%

          \[\leadsto 1 - \frac{1}{2 + \color{blue}{t \cdot \left(4 + \frac{-2}{1 + t} \cdot 2\right)}} \]

       8. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \frac{\color{blue}{-2}}{1 + t} \cdot 2\right)} \]

       9. distribute-neg-frac 100.0%

          \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \color{blue}{\left(-\frac{2}{1 + t}\right)} \cdot 2\right)} \]

       10. distribute-neg-frac2 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \color{blue}{\frac{2}{-\left(1 + t\right)}} \cdot 2\right)} \]

       11. associate-*l/ 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \color{blue}{\frac{2 \cdot 2}{-\left(1 + t\right)}}\right)} \]

       12. metadata-eval 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \frac{\color{blue}{4}}{-\left(1 + t\right)}\right)} \]

       13. distribute-neg-in 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \frac{4}{\color{blue}{\left(-1\right) + \left(-t\right)}}\right)} \]

       14. metadata-eval 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \frac{4}{\color{blue}{-1} + \left(-t\right)}\right)} \]

       15. unsub-neg 100.0%

           \[\leadsto 1 - \frac{1}{2 + t \cdot \left(4 + \frac{4}{\color{blue}{-1 - t}}\right)} \]

   17. Simplified 100.0%

       \[\leadsto 1 - \frac{1}{2 + \color{blue}{t \cdot \left(4 + \frac{4}{-1 - t}\right)}} \]

   if 1.55000000000000004 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{4}{1 + t} - 4\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.64:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 1.2:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.64)
   (-
    1.0
    (+
     (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)
     0.16666666666666666))
   (if (<= t 1.2)
     (+ 1.0 (/ -1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (+ 1.0 (/ -1.0 (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.64) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.2) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.64d0)) then
        tmp = 1.0d0 - (((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t) + 0.16666666666666666d0)
    else if (t <= 1.2d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.64) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.2) {
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.64:
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666)
	elif t <= 1.2:
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.64)
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t) + 0.16666666666666666));
	elseif (t <= 1.2)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.64)
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	elseif (t <= 1.2)
		tmp = 1.0 + (-1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.64], N[(1.0 - N[(N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2], N[(1.0 + N[(-1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.640000000000000013

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - \frac{0.037037037037037035}{{t}^{2}}\right)} \]
   4. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto 1 - \color{blue}{\left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) + 0.16666666666666666\right)} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right) + 0.16666666666666666\right) \]

      4. associate-/r* 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right) + 0.16666666666666666\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t}\right) + 0.16666666666666666\right) \]

      6. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t}\right) + 0.16666666666666666\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      8. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      9. div-sub 100.0%

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} + 0.16666666666666666\right) \]

      10. sub-neg 100.0%

          \[\leadsto 1 - \left(\frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} + 0.16666666666666666\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} + 0.16666666666666666\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} + 0.16666666666666666\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} + 0.16666666666666666\right) \]

      14. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} + 0.16666666666666666\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)} \]

   if -0.640000000000000013 < t < 1.19999999999999996

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. div-inv 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right)} \]
   5. Step-by-step derivation

      1. unpow-1 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

      4. associate-*l* 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

      5. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

      6. *-rgt-identity 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

      7. rgt-mult-inverse 100.0%

         \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   6. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(2 \cdot t\right)}} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

      \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \color{blue}{\left(t \cdot 2\right)}} \]
   10. Step-by-step derivation

       1. clear-num 100.0%

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

       2. inv-pow 100.0%

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

       3. div-inv 100.0%

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

       4. clear-num 100.0%

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

       5. div-inv 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - {\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot \color{blue}{0.5}\right)\right)}^{-1}\right)} \]

   11. Applied egg-rr 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{{\left(\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)\right)}^{-1}}\right) \cdot \left(t \cdot 2\right)} \]
   12. Step-by-step derivation

       1. unpow-1 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \color{blue}{\frac{1}{\left(1 + \frac{1}{t}\right) \cdot \left(t \cdot 0.5\right)}}\right)} \]

       2. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(t \cdot 0.5\right) \cdot \left(1 + \frac{1}{t}\right)}}\right)} \]

       3. *-commutative 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{\left(0.5 \cdot t\right)} \cdot \left(1 + \frac{1}{t}\right)}\right)} \]

       4. associate-*l* 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{\color{blue}{0.5 \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}}\right)} \]

       5. distribute-lft-in 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \color{blue}{\left(t \cdot 1 + t \cdot \frac{1}{t}\right)}}\right)} \]

       6. *-rgt-identity 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(\color{blue}{t} + t \cdot \frac{1}{t}\right)}\right)} \]

       7. rgt-mult-inverse 100.0%

          \[\leadsto 1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{1}{0.5 \cdot \left(t + \color{blue}{1}\right)}\right)} \]

   13. Simplified 100.0%

       \[\leadsto 1 - \frac{1}{2 + \left(2 - \color{blue}{\frac{1}{0.5 \cdot \left(t + 1\right)}}\right) \cdot \left(t \cdot 2\right)} \]

   14. Taylor expanded in t around 0 100.0%

       \[\leadsto 1 - \frac{1}{2 + \color{blue}{\left(2 \cdot t\right)} \cdot \left(t \cdot 2\right)} \]
   15. Step-by-step derivation

       1. *-commutative 100.0%

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

   16. Simplified 100.0%

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

   if 1.19999999999999996 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.64:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 1.2:\\ \;\;\;\;1 + \frac{-1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 1.16:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.52)
   (-
    1.0
    (+
     (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)
     0.16666666666666666))
   (if (<= t 1.16) 0.5 (+ 1.0 (/ -1.0 (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.16) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.52d0)) then
        tmp = 1.0d0 - (((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t) + 0.16666666666666666d0)
    else if (t <= 1.16d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	} else if (t <= 1.16) {
		tmp = 0.5;
	} else {
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.52:
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666)
	elif t <= 1.16:
		tmp = 0.5
	else:
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(1.0 - Float64(Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t) + 0.16666666666666666));
	elseif (t <= 1.16)
		tmp = 0.5;
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 1.0 - (((0.2222222222222222 + (-0.037037037037037035 / t)) / t) + 0.16666666666666666);
	elseif (t <= 1.16)
		tmp = 0.5;
	else
		tmp = 1.0 + (-1.0 / (6.0 - ((8.0 + (-12.0 / t)) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.52], N[(1.0 - N[(N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16], 0.5, N[(1.0 + N[(-1.0 / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - \frac{0.037037037037037035}{{t}^{2}}\right)} \]
   4. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto 1 - \color{blue}{\left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) + 0.16666666666666666\right)} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right) + 0.16666666666666666\right) \]

      4. associate-/r* 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right) + 0.16666666666666666\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t}\right) + 0.16666666666666666\right) \]

      6. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t}\right) + 0.16666666666666666\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      8. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      9. div-sub 100.0%

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} + 0.16666666666666666\right) \]

      10. sub-neg 100.0%

          \[\leadsto 1 - \left(\frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} + 0.16666666666666666\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} + 0.16666666666666666\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} + 0.16666666666666666\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} + 0.16666666666666666\right) \]

      14. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} + 0.16666666666666666\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)} \]

   if -0.52000000000000002 < t < 1.15999999999999992

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]

   if 1.15999999999999992 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto 1 + \frac{-1}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.52) (not (<= t 0.23)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.52d0)) .or. (.not. (t <= 0.23d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.52) || !(t <= 0.23)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.52) or not (t <= 0.23):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.52) || !(t <= 0.23))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.52) || ~((t <= 0.23)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.52], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 99.9%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}\right)} \]

      2. unsub-neg 99.9%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

      3. sub-neg 99.9%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} \]

      4. associate-*r/ 99.9%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} \]

      5. metadata-eval 99.9%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} \]

      6. distribute-neg-frac 99.9%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} \]

      7. metadata-eval 99.9%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}} \]

   if -0.52000000000000002 < t < 0.23000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52 \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;1 - \left(t\_1 + 0.16666666666666666\right)\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))
   (if (<= t -0.52)
     (- 1.0 (+ t_1 0.16666666666666666))
     (if (<= t 0.23) 0.5 (- 0.8333333333333334 t_1)))))

C

double code(double t) {
	double t_1 = (0.2222222222222222 + (-0.037037037037037035 / t)) / t;
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 - (t_1 + 0.16666666666666666);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t
    if (t <= (-0.52d0)) then
        tmp = 1.0d0 - (t_1 + 0.16666666666666666d0)
    else if (t <= 0.23d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (0.2222222222222222 + (-0.037037037037037035 / t)) / t;
	double tmp;
	if (t <= -0.52) {
		tmp = 1.0 - (t_1 + 0.16666666666666666);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (0.2222222222222222 + (-0.037037037037037035 / t)) / t
	tmp = 0
	if t <= -0.52:
		tmp = 1.0 - (t_1 + 0.16666666666666666)
	elif t <= 0.23:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t)
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(1.0 - Float64(t_1 + 0.16666666666666666));
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (0.2222222222222222 + (-0.037037037037037035 / t)) / t;
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 1.0 - (t_1 + 0.16666666666666666);
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -0.52], N[(1.0 - N[(t$95$1 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.23], 0.5, N[(0.8333333333333334 - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 - \color{blue}{\left(\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right) - \frac{0.037037037037037035}{{t}^{2}}\right)} \]
   4. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto 1 - \color{blue}{\left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) + 0.16666666666666666\right)} \]

      3. unpow2 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{\color{blue}{t \cdot t}}\right) + 0.16666666666666666\right) \]

      4. associate-/r* 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \color{blue}{\frac{\frac{0.037037037037037035}{t}}{t}}\right) + 0.16666666666666666\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\frac{\color{blue}{0.037037037037037035 \cdot 1}}{t}}{t}\right) + 0.16666666666666666\right) \]

      6. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(0.2222222222222222 \cdot \frac{1}{t} - \frac{\color{blue}{0.037037037037037035 \cdot \frac{1}{t}}}{t}\right) + 0.16666666666666666\right) \]

      7. associate-*r/ 100.0%

         \[\leadsto 1 - \left(\left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      8. metadata-eval 100.0%

         \[\leadsto 1 - \left(\left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035 \cdot \frac{1}{t}}{t}\right) + 0.16666666666666666\right) \]

      9. div-sub 100.0%

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} + 0.16666666666666666\right) \]

      10. sub-neg 100.0%

          \[\leadsto 1 - \left(\frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} + 0.16666666666666666\right) \]

      11. associate-*r/ 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} + 0.16666666666666666\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} + 0.16666666666666666\right) \]

      13. distribute-neg-frac 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} + 0.16666666666666666\right) \]

      14. metadata-eval 100.0%

          \[\leadsto 1 - \left(\frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} + 0.16666666666666666\right) \]

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t} + 0.16666666666666666\right)} \]

   if -0.52000000000000002 < t < 0.23000000000000001

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]

   if 0.23000000000000001 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 0.8333333333333334 + \color{blue}{\left(-\frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - 0.037037037037037035 \cdot \frac{1}{t}}{t}} \]

      3. sub-neg 100.0%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222 + \left(-0.037037037037037035 \cdot \frac{1}{t}\right)}}{t} \]

      4. associate-*r/ 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \left(-\color{blue}{\frac{0.037037037037037035 \cdot 1}{t}}\right)}{t} \]

      5. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \left(-\frac{\color{blue}{0.037037037037037035}}{t}\right)}{t} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \color{blue}{\frac{-0.037037037037037035}{t}}}{t} \]

      7. metadata-eval 100.0%

         \[\leadsto 0.8333333333333334 - \frac{0.2222222222222222 + \frac{\color{blue}{-0.037037037037037035}}{t}}{t} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.660000000000000031 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.7%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.7%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.49)
   (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))
   (if (<= t 0.66) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.49d0)) then
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    else if (t <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.49) {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.49:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.49)
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.49)
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.49], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.66], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.48999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 99.7%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto 1 - \color{blue}{\left(0.2222222222222222 \cdot \frac{1}{t} + 0.16666666666666666\right)} \]

      2. associate-*r/ 99.7%

         \[\leadsto 1 - \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} + 0.16666666666666666\right) \]

      3. metadata-eval 99.7%

         \[\leadsto 1 - \left(\frac{\color{blue}{0.2222222222222222}}{t} + 0.16666666666666666\right) \]

   5. Simplified 99.7%

      \[\leadsto 1 - \color{blue}{\left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)} \]

   if -0.48999999999999999 < t < 0.660000000000000031

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]

   if 0.660000000000000031 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 99.7%

      \[\leadsto \color{blue}{0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} \]

      2. metadata-eval 99.7%

         \[\leadsto 0.8333333333333334 - \frac{\color{blue}{0.2222222222222222}}{t} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 97.6%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

      \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. distribute-neg-frac 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

         \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-neg-frac 100.0%

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

   3. metadata-eval 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

      \[\leadsto 1 + \frac{-1}{\color{blue}{\mathsf{fma}\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}, 2\right)}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 + \frac{-1}{\mathsf{fma}\left(2 - \frac{-2}{-1 - t}, 2 - \frac{-2}{-1 - t}, 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 60.9%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(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)))))))))