Kahan p13 Example 2

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \frac{1}{t}\\ t_2 := 2 - \frac{\frac{2}{t}}{t\_1}\\ t_3 := 2 + \frac{\frac{-2}{t}}{t\_1}\\ \frac{1 + t\_2 \cdot t\_2}{\frac{{t\_3}^{4} - 4}{t\_3 \cdot t\_3 - 2}} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (/ 1.0 t)))
        (t_2 (- 2.0 (/ (/ 2.0 t) t_1)))
        (t_3 (+ 2.0 (/ (/ -2.0 t) t_1))))
   (/ (+ 1.0 (* t_2 t_2)) (/ (- (pow t_3 4.0) 4.0) (- (* t_3 t_3) 2.0)))))

C

double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = 2.0 - ((2.0 / t) / t_1);
	double t_3 = 2.0 + ((-2.0 / t) / t_1);
	return (1.0 + (t_2 * t_2)) / ((pow(t_3, 4.0) - 4.0) / ((t_3 * t_3) - 2.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_1 = 1.0d0 + (1.0d0 / t)
    t_2 = 2.0d0 - ((2.0d0 / t) / t_1)
    t_3 = 2.0d0 + (((-2.0d0) / t) / t_1)
    code = (1.0d0 + (t_2 * t_2)) / (((t_3 ** 4.0d0) - 4.0d0) / ((t_3 * t_3) - 2.0d0))
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (1.0 / t);
	double t_2 = 2.0 - ((2.0 / t) / t_1);
	double t_3 = 2.0 + ((-2.0 / t) / t_1);
	return (1.0 + (t_2 * t_2)) / ((Math.pow(t_3, 4.0) - 4.0) / ((t_3 * t_3) - 2.0));
}

Python

def code(t):
	t_1 = 1.0 + (1.0 / t)
	t_2 = 2.0 - ((2.0 / t) / t_1)
	t_3 = 2.0 + ((-2.0 / t) / t_1)
	return (1.0 + (t_2 * t_2)) / ((math.pow(t_3, 4.0) - 4.0) / ((t_3 * t_3) - 2.0))

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(1.0 / t))
	t_2 = Float64(2.0 - Float64(Float64(2.0 / t) / t_1))
	t_3 = Float64(2.0 + Float64(Float64(-2.0 / t) / t_1))
	return Float64(Float64(1.0 + Float64(t_2 * t_2)) / Float64(Float64((t_3 ^ 4.0) - 4.0) / Float64(Float64(t_3 * t_3) - 2.0)))
end

MATLAB

function tmp = code(t)
	t_1 = 1.0 + (1.0 / t);
	t_2 = 2.0 - ((2.0 / t) / t_1);
	t_3 = 2.0 + ((-2.0 / t) / t_1);
	tmp = (1.0 + (t_2 * t_2)) / (((t_3 ^ 4.0) - 4.0) / ((t_3 * t_3) - 2.0));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[(-2.0 / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$3, 4.0], $MachinePrecision] - 4.0), $MachinePrecision] / N[(N[(t$95$3 * t$95$3), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. flip-+ 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;t \leq -0.45:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.72:\\ \;\;\;\;\frac{1 + t\_1 \cdot \left(t \cdot \left(2 + t \cdot \left(2 \cdot t - 2\right)\right)\right)}{2 + t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= t -0.45)
     0.8333333333333334
     (if (<= t 0.72)
       (/
        (+ 1.0 (* t_1 (* t (+ 2.0 (* t (- (* 2.0 t) 2.0))))))
        (+ 2.0 (* t_1 t_1)))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if (t <= -0.45) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.72) {
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * t_1));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    if (t <= (-0.45d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.72d0) then
        tmp = (1.0d0 + (t_1 * (t * (2.0d0 + (t * ((2.0d0 * t) - 2.0d0)))))) / (2.0d0 + (t_1 * t_1))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if (t <= -0.45) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.72) {
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * t_1));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	tmp = 0
	if t <= -0.45:
		tmp = 0.8333333333333334
	elif t <= 0.72:
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * t_1))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (t <= -0.45)
		tmp = 0.8333333333333334;
	elseif (t <= 0.72)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(t * Float64(2.0 + Float64(t * Float64(Float64(2.0 * t) - 2.0)))))) / Float64(2.0 + Float64(t_1 * t_1)));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 0.0;
	if (t <= -0.45)
		tmp = 0.8333333333333334;
	elseif (t <= 0.72)
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * t_1));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
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]}, If[LessEqual[t, -0.45], 0.8333333333333334, If[LessEqual[t, 0.72], N[(N[(1.0 + N[(t$95$1 * N[(t * N[(2.0 + N[(t * N[(N[(2.0 * t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.450000000000000011

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.450000000000000011 < t < 0.71999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   if 0.71999999999999997 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \mathbf{if}\;t \leq -0.455:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;\frac{1 + t\_1 \cdot \left(t \cdot \left(2 + t \cdot \left(2 \cdot t - 2\right)\right)\right)}{2 + t\_1 \cdot \left(t \cdot \left(2 + -2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (if (<= t -0.455)
     0.8333333333333334
     (if (<= t 0.65)
       (/
        (+ 1.0 (* t_1 (* t (+ 2.0 (* t (- (* 2.0 t) 2.0))))))
        (+ 2.0 (* t_1 (* t (+ 2.0 (* -2.0 t))))))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if (t <= -0.455) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * (t * (2.0 + (-2.0 * t)))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    if (t <= (-0.455d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.65d0) then
        tmp = (1.0d0 + (t_1 * (t * (2.0d0 + (t * ((2.0d0 * t) - 2.0d0)))))) / (2.0d0 + (t_1 * (t * (2.0d0 + ((-2.0d0) * t)))))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double tmp;
	if (t <= -0.455) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * (t * (2.0 + (-2.0 * t)))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	tmp = 0
	if t <= -0.455:
		tmp = 0.8333333333333334
	elif t <= 0.65:
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * (t * (2.0 + (-2.0 * t)))))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	tmp = 0.0
	if (t <= -0.455)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = Float64(Float64(1.0 + Float64(t_1 * Float64(t * Float64(2.0 + Float64(t * Float64(Float64(2.0 * t) - 2.0)))))) / Float64(2.0 + Float64(t_1 * Float64(t * Float64(2.0 + Float64(-2.0 * t))))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = 0.0;
	if (t <= -0.455)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = (1.0 + (t_1 * (t * (2.0 + (t * ((2.0 * t) - 2.0)))))) / (2.0 + (t_1 * (t * (2.0 + (-2.0 * t)))));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
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]}, If[LessEqual[t, -0.455], 0.8333333333333334, If[LessEqual[t, 0.65], N[(N[(1.0 + N[(t$95$1 * N[(t * N[(2.0 + N[(t * N[(N[(2.0 * t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.455000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.455000000000000016 < t < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.8%

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

   4. Taylor expanded in t around 0 99.6%

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

   if 0.650000000000000022 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
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]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

Alternative 5: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(2 + -2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;\frac{1 + t\_1 \cdot t\_1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ 2.0 (* -2.0 t)))))
   (if (<= t -0.62)
     0.8333333333333334
     (if (<= t 0.58)
       (/
        (+ 1.0 (* t_1 t_1))
        (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (* 2.0 t))))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.62) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (2.0d0 + ((-2.0d0) * t))
    if (t <= (-0.62d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.58d0) then
        tmp = (1.0d0 + (t_1 * t_1)) / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 * t)))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.62) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.58) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (2.0 + (-2.0 * t))
	tmp = 0
	if t <= -0.62:
		tmp = 0.8333333333333334
	elif t <= 0.58:
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(2.0 + Float64(-2.0 * t)))
	tmp = 0.0
	if (t <= -0.62)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = Float64(Float64(1.0 + Float64(t_1 * t_1)) / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (2.0 + (-2.0 * t));
	tmp = 0.0;
	if (t <= -0.62)
		tmp = 0.8333333333333334;
	elseif (t <= 0.58)
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.62], 0.8333333333333334, If[LessEqual[t, 0.58], N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.619999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.619999999999999996 < t < 0.57999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

   5. Taylor expanded in t around 0 99.4%

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

   if 0.57999999999999996 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(2 + -2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.44:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.6:\\ \;\;\;\;\frac{1 + t\_1 \cdot t\_1}{2 + t\_1 \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ 2.0 (* -2.0 t)))))
   (if (<= t -0.44)
     0.8333333333333334
     (if (<= t 0.6)
       (/ (+ 1.0 (* t_1 t_1)) (+ 2.0 (* t_1 (* 2.0 t))))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.44) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (2.0d0 + ((-2.0d0) * t))
    if (t <= (-0.44d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.6d0) then
        tmp = (1.0d0 + (t_1 * t_1)) / (2.0d0 + (t_1 * (2.0d0 * t)))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.44) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.6) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (2.0 + (-2.0 * t))
	tmp = 0
	if t <= -0.44:
		tmp = 0.8333333333333334
	elif t <= 0.6:
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 * t)))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(2.0 + Float64(-2.0 * t)))
	tmp = 0.0
	if (t <= -0.44)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		tmp = Float64(Float64(1.0 + Float64(t_1 * t_1)) / Float64(2.0 + Float64(t_1 * Float64(2.0 * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (2.0 + (-2.0 * t));
	tmp = 0.0;
	if (t <= -0.44)
		tmp = 0.8333333333333334;
	elseif (t <= 0.6)
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + (t_1 * (2.0 * t)));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.44], 0.8333333333333334, If[LessEqual[t, 0.6], N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t$95$1 * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.440000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.440000000000000002 < t < 0.599999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

   5. Taylor expanded in t around 0 99.4%

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

   6. Taylor expanded in t around 0 99.4%

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

   if 0.599999999999999978 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;\frac{1 + \left(2 \cdot t\right) \cdot \left(t \cdot \left(2 + -2 \cdot t\right)\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.39)
   0.8333333333333334
   (if (<= t 0.7)
     (/
      (+ 1.0 (* (* 2.0 t) (* t (+ 2.0 (* -2.0 t)))))
      (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (* 2.0 t))))
     (+
      0.8333333333333334
      (*
       -1.0
       (/
        (+
         0.2222222222222222
         (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.7) {
		tmp = (1.0 + ((2.0 * t) * (t * (2.0 + (-2.0 * t))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.39d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.7d0) then
        tmp = (1.0d0 + ((2.0d0 * t) * (t * (2.0d0 + ((-2.0d0) * t))))) / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (2.0d0 * t)))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.7) {
		tmp = (1.0 + ((2.0 * t) * (t * (2.0 + (-2.0 * t))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.39:
		tmp = 0.8333333333333334
	elif t <= 0.7:
		tmp = (1.0 + ((2.0 * t) * (t * (2.0 + (-2.0 * t))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.7)
		tmp = Float64(Float64(1.0 + Float64(Float64(2.0 * t) * Float64(t * Float64(2.0 + Float64(-2.0 * t))))) / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(2.0 * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.7)
		tmp = (1.0 + ((2.0 * t) * (t * (2.0 + (-2.0 * t))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 * t)));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.39], 0.8333333333333334, If[LessEqual[t, 0.7], N[(N[(1.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.39000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.39000000000000001 < t < 0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

   5. Taylor expanded in t around 0 99.4%

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

   if 0.69999999999999996 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(2 + -2 \cdot t\right)\\ \mathbf{if}\;t \leq -0.39:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.52:\\ \;\;\;\;\frac{1 + t\_1 \cdot t\_1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ 2.0 (* -2.0 t)))))
   (if (<= t -0.39)
     0.8333333333333334
     (if (<= t 0.52)
       (/ (+ 1.0 (* t_1 t_1)) (+ 2.0 (* (* 2.0 t) (* 2.0 t))))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.52) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 * t) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (2.0d0 + ((-2.0d0) * t))
    if (t <= (-0.39d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.52d0) then
        tmp = (1.0d0 + (t_1 * t_1)) / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t)))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -0.39) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.52) {
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 * t) * (2.0 * t)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (2.0 + (-2.0 * t))
	tmp = 0
	if t <= -0.39:
		tmp = 0.8333333333333334
	elif t <= 0.52:
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 * t) * (2.0 * t)))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(2.0 + Float64(-2.0 * t)))
	tmp = 0.0
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.52)
		tmp = Float64(Float64(1.0 + Float64(t_1 * t_1)) / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (2.0 + (-2.0 * t));
	tmp = 0.0;
	if (t <= -0.39)
		tmp = 0.8333333333333334;
	elseif (t <= 0.52)
		tmp = (1.0 + (t_1 * t_1)) / (2.0 + ((2.0 * t) * (2.0 * t)));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.39], 0.8333333333333334, If[LessEqual[t, 0.52], N[(N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.39000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.39000000000000001 < t < 0.52000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

   5. Taylor expanded in t around 0 99.4%

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

   6. Taylor expanded in t around 0 99.3%

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

   if 0.52000000000000002 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.66)
     0.5
     (+
      0.8333333333333334
      (*
       -1.0
       (/
        (+
         0.2222222222222222
         (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	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 <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    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 <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 0.66], 0.5, N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.23)
     0.5
     (+
      0.8333333333333334
      (/ (- (/ 0.037037037037037035 t) 0.2222222222222222) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	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 <= 0.23d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + (((0.037037037037037035d0 / t) - 0.2222222222222222d0) / t)
    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 <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 0.23:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(0.037037037037037035 / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + (((0.037037037037037035 / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 0.23], 0.5, N[(0.8333333333333334 + N[(N[(N[(0.037037037037037035 / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

   if 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]

   4. Taylor expanded in t around inf 98.1%

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

   5. Taylor expanded in t around 0 98.1%

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

Alternative 11: 99.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.66) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} 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 <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    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 <= -0.33) {
		tmp = 0.8333333333333334;
	} 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 <= -0.33:
		tmp = 0.8333333333333334
	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 <= -0.33)
		tmp = 0.8333333333333334;
	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 <= -0.33)
		tmp = 0.8333333333333334;
	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, -0.33], 0.8333333333333334, 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 < -0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.5%

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

   4. Taylor expanded in t around 0 97.5%

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

Alternative 12: 98.7% accurate, 4.6× 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%

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

   3. Taylor expanded in t around inf 98.0%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

Alternative 13: 59.1% accurate, 51.0× 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%

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

3. Taylor expanded in t around 0 59.4%

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

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))