Kahan p13 Example 2

Percentage Accurate: 100.0% → 100.0%

Time: 16.0s

Alternatives: 12

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: 100.0% 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: 100.0% 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 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ t_2 := 4 + t \cdot \left(-8 + t \cdot \left(12 + t \cdot -16\right)\right)\\ \mathbf{if}\;t \leq -0.9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.62:\\ \;\;\;\;\frac{1 + \left(t \cdot t\right) \cdot t\_2}{2 + t \cdot \left(t \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (+
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            -0.2222222222222222)
           t)))
        (t_2 (+ 4.0 (* t (+ -8.0 (* t (+ 12.0 (* t -16.0))))))))
   (if (<= t -0.9)
     t_1
     (if (<= t 0.62)
       (/ (+ 1.0 (* (* t t) t_2)) (+ 2.0 (* t (* t t_2))))
       t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double t_2 = 4.0 + (t * (-8.0 + (t * (12.0 + (t * -16.0)))));
	double tmp;
	if (t <= -0.9) {
		tmp = t_1;
	} else if (t <= 0.62) {
		tmp = (1.0 + ((t * t) * t_2)) / (2.0 + (t * (t * t_2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) + (-0.2222222222222222d0)) / t)
    t_2 = 4.0d0 + (t * ((-8.0d0) + (t * (12.0d0 + (t * (-16.0d0))))))
    if (t <= (-0.9d0)) then
        tmp = t_1
    else if (t <= 0.62d0) then
        tmp = (1.0d0 + ((t * t) * t_2)) / (2.0d0 + (t * (t * t_2)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double t_2 = 4.0 + (t * (-8.0 + (t * (12.0 + (t * -16.0)))));
	double tmp;
	if (t <= -0.9) {
		tmp = t_1;
	} else if (t <= 0.62) {
		tmp = (1.0 + ((t * t) * t_2)) / (2.0 + (t * (t * t_2)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t)
	t_2 = 4.0 + (t * (-8.0 + (t * (12.0 + (t * -16.0)))))
	tmp = 0
	if t <= -0.9:
		tmp = t_1
	elif t <= 0.62:
		tmp = (1.0 + ((t * t) * t_2)) / (2.0 + (t * (t * t_2)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t))
	t_2 = Float64(4.0 + Float64(t * Float64(-8.0 + Float64(t * Float64(12.0 + Float64(t * -16.0))))))
	tmp = 0.0
	if (t <= -0.9)
		tmp = t_1;
	elseif (t <= 0.62)
		tmp = Float64(Float64(1.0 + Float64(Float64(t * t) * t_2)) / Float64(2.0 + Float64(t * Float64(t * t_2))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	t_2 = 4.0 + (t * (-8.0 + (t * (12.0 + (t * -16.0)))));
	tmp = 0.0;
	if (t <= -0.9)
		tmp = t_1;
	elseif (t <= 0.62)
		tmp = (1.0 + ((t * t) * t_2)) / (2.0 + (t * (t * t_2)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 + N[(t * N[(-8.0 + N[(t * N[(12.0 + N[(t * -16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.9], t$95$1, If[LessEqual[t, 0.62], N[(N[(1.0 + N[(N[(t * t), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(t * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.900000000000000022 or 0.619999999999999996 < 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

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}\right), \color{blue}{t}\right)\right) \]

   5. Simplified 99.8%

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

   if -0.900000000000000022 < 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 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \left(\left(t \cdot t\right) \cdot \left(\color{blue}{4} + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \left(t \cdot \color{blue}{\left(t \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \left(t \cdot \left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot \color{blue}{t}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right) \cdot t\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \left(t \cdot \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \color{blue}{\left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)}\right)\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(t \cdot \left(12 + -16 \cdot t\right) + \color{blue}{\left(\mathsf{neg}\left(8\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(t \cdot \left(12 + -16 \cdot t\right) + -8\right)\right)\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(-8 + \color{blue}{t \cdot \left(12 + -16 \cdot t\right)}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \color{blue}{\left(t \cdot \left(12 + -16 \cdot t\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \color{blue}{\left(12 + -16 \cdot t\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \color{blue}{\left(-16 \cdot t\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \left(t \cdot \color{blue}{-16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, \color{blue}{-16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \color{blue}{t \cdot \left(t \cdot \left(4 + t \cdot \left(-8 + t \cdot \left(12 + t \cdot -16\right)\right)\right)\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\left({t}^{2} \cdot \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(4 + t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \left(t \cdot \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(t \cdot \left(12 + -16 \cdot t\right) - 8\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(t \cdot \left(12 + -16 \cdot t\right) + \left(\mathsf{neg}\left(8\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(t \cdot \left(12 + -16 \cdot t\right) + -8\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \left(-8 + t \cdot \left(12 + -16 \cdot t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \left(t \cdot \left(12 + -16 \cdot t\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \left(12 + -16 \cdot t\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \left(-16 \cdot t\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \left(t \cdot -16\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-8, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(12, \mathsf{*.f64}\left(t, -16\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto \frac{1 + \color{blue}{\left(t \cdot t\right) \cdot \left(4 + t \cdot \left(-8 + t \cdot \left(12 + t \cdot -16\right)\right)\right)}}{2 + t \cdot \left(t \cdot \left(4 + t \cdot \left(-8 + t \cdot \left(12 + t \cdot -16\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{4}{-1 - t}\\ t_2 := 2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\\ \frac{1 + t\_2 \cdot t\_2}{\frac{4 + t\_1}{-1 - t} + \left(t\_1 + 6\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 4.0 (- -1.0 t))) (t_2 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
   (/ (+ 1.0 (* t_2 t_2)) (+ (/ (+ 4.0 t_1) (- -1.0 t)) (+ t_1 6.0)))))

C

double code(double t) {
	double t_1 = 4.0 / (-1.0 - t);
	double t_2 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return (1.0 + (t_2 * t_2)) / (((4.0 + t_1) / (-1.0 - t)) + (t_1 + 6.0));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 4.0d0 / ((-1.0d0) - t)
    t_2 = 2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))
    code = (1.0d0 + (t_2 * t_2)) / (((4.0d0 + t_1) / ((-1.0d0) - t)) + (t_1 + 6.0d0))
end function

Java

public static double code(double t) {
	double t_1 = 4.0 / (-1.0 - t);
	double t_2 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	return (1.0 + (t_2 * t_2)) / (((4.0 + t_1) / (-1.0 - t)) + (t_1 + 6.0));
}

Python

def code(t):
	t_1 = 4.0 / (-1.0 - t)
	t_2 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))
	return (1.0 + (t_2 * t_2)) / (((4.0 + t_1) / (-1.0 - t)) + (t_1 + 6.0))

Julia

function code(t)
	t_1 = Float64(4.0 / Float64(-1.0 - t))
	t_2 = Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))))
	return Float64(Float64(1.0 + Float64(t_2 * t_2)) / Float64(Float64(Float64(4.0 + t_1) / Float64(-1.0 - t)) + Float64(t_1 + 6.0)))
end

MATLAB

function tmp = code(t)
	t_1 = 4.0 / (-1.0 - t);
	t_2 = 2.0 - ((2.0 / t) / (1.0 + (1.0 / t)));
	tmp = (1.0 + (t_2 * t_2)) / (((4.0 + t_1) / (-1.0 - t)) + (t_1 + 6.0));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 + t$95$1), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + 6.0), $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

4. Applied egg-rr 99.8%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \left(\left(\frac{\frac{\frac{4}{t}}{1 + \frac{1}{t}} - 4}{t \cdot \left(1 + \frac{1}{t}\right)} + 2\right) + \color{blue}{\left(4 + \frac{4}{t \cdot \left(-1 + \frac{-1}{t}\right)}\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{\frac{4}{t}}{1 + \frac{1}{t}} - 4}{t \cdot \left(1 + \frac{1}{t}\right)} + 2\right), \color{blue}{\left(4 + \frac{4}{t \cdot \left(-1 + \frac{-1}{t}\right)}\right)}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\left(\frac{\frac{4}{t + 1} + -4}{t + 1} + 2\right) + \left(4 + \frac{4}{-1 + \left(0 - t\right)}\right)}} \]
7. Step-by-step derivation

   1. associate-+r- N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{+.f64}\left(t, 1\right)\right), -4\right), \mathsf{+.f64}\left(t, 1\right)\right), 2\right), \mathsf{+.f64}\left(4, \mathsf{/.f64}\left(4, \left(\left(-1 + 0\right) - \color{blue}{t}\right)\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{+.f64}\left(t, 1\right)\right), -4\right), \mathsf{+.f64}\left(t, 1\right)\right), 2\right), \mathsf{+.f64}\left(4, \mathsf{/.f64}\left(4, \left(-1 - t\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(4, \mathsf{+.f64}\left(t, 1\right)\right), -4\right), \mathsf{+.f64}\left(t, 1\right)\right), 2\right), \mathsf{+.f64}\left(4, \mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(-1, \color{blue}{t}\right)\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

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

   1. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \left(\frac{\frac{4}{t + 1} + -4}{t + 1} + \color{blue}{\left(2 + \left(4 + \frac{4}{-1 - t}\right)\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, t\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, t\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{\frac{4}{t + 1} + -4}{t + 1}\right), \color{blue}{\left(2 + \left(4 + \frac{4}{-1 - t}\right)\right)}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\color{blue}{\frac{4 + \frac{4}{-1 - t}}{-1 - t} + \left(6 + \frac{4}{-1 - t}\right)}} \]

11. Final simplification 100.0%

    \[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{\frac{4 + \frac{4}{-1 - t}}{-1 - t} + \left(\frac{4}{-1 - t} + 6\right)} \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right)\right)\right) + \left(t \cdot t + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (+
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            -0.2222222222222222)
           t))))
   (if (<= t -0.35)
     t_1
     (if (<= t 0.74) (+ (* t (* t (* t (+ t -2.0)))) (+ (* t t) 0.5)) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.74) {
		tmp = (t * (t * (t * (t + -2.0)))) + ((t * t) + 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) + (-0.2222222222222222d0)) / t)
    if (t <= (-0.35d0)) then
        tmp = t_1
    else if (t <= 0.74d0) then
        tmp = (t * (t * (t * (t + (-2.0d0))))) + ((t * t) + 0.5d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.74) {
		tmp = (t * (t * (t * (t + -2.0)))) + ((t * t) + 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t)
	tmp = 0
	if t <= -0.35:
		tmp = t_1
	elif t <= 0.74:
		tmp = (t * (t * (t * (t + -2.0)))) + ((t * t) + 0.5)
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.74)
		tmp = Float64(Float64(t * Float64(t * Float64(t * Float64(t + -2.0)))) + Float64(Float64(t * t) + 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.74)
		tmp = (t * (t * (t * (t + -2.0)))) + ((t * t) + 0.5);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.35], t$95$1, If[LessEqual[t, 0.74], N[(N[(t * N[(t * N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.34999999999999998 or 0.73999999999999999 < 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

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}\right), \color{blue}{t}\right)\right) \]

   5. Simplified 99.8%

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

   if -0.34999999999999998 < t < 0.73999999999999999

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(-2 + \color{blue}{t}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-2, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} + t \cdot t\right), \color{blue}{\left(\left(t \cdot t\right) \cdot \left(t \cdot \left(-2 + t\right)\right)\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right), \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(-2 + t\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \left(\left(t \cdot \color{blue}{t}\right) \cdot \left(t \cdot \left(-2 + t\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \left(t \cdot \color{blue}{\left(t \cdot \left(t \cdot \left(-2 + t\right)\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(t \cdot \left(-2 + t\right)\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(-2 + t\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(-2 + t\right)}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(t + \color{blue}{-2}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(t, \color{blue}{-2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(0.5 + t \cdot t\right) + t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;t \cdot \left(t \cdot \left(t \cdot \left(t + -2\right)\right)\right) + \left(t \cdot t + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/
           (+
            (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
            -0.2222222222222222)
           t))))
   (if (<= t -0.35)
     t_1
     (if (<= t 0.74) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.74) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) + (-0.2222222222222222d0)) / t)
    if (t <= (-0.35d0)) then
        tmp = t_1
    else if (t <= 0.74d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	double tmp;
	if (t <= -0.35) {
		tmp = t_1;
	} else if (t <= 0.74) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t)
	tmp = 0
	if t <= -0.35:
		tmp = t_1
	elif t <= 0.74:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) + -0.2222222222222222) / t))
	tmp = 0.0
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.74)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) + -0.2222222222222222) / t);
	tmp = 0.0;
	if (t <= -0.35)
		tmp = t_1;
	elseif (t <= 0.74)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + -0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.35], t$95$1, If[LessEqual[t, 0.74], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.34999999999999998 or 0.73999999999999999 < 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

      \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)}{\color{blue}{t}}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\left(\frac{2}{9} + -1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)}{t}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}\right)\right)\right)\right)}{t}\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\left(\mathsf{neg}\left(\frac{2}{9}\right)\right) + \frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t}}{t}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} + \left(\mathsf{neg}\left(\frac{2}{9}\right)\right)}{t}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}}{t}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{27} + \frac{4}{81} \cdot \frac{1}{t}}{t} - \frac{2}{9}\right), \color{blue}{t}\right)\right) \]

   5. Simplified 99.8%

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

   if -0.34999999999999998 < t < 0.73999999999999999

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(-2 + \color{blue}{t}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-2, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(-2 + t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.74:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} + -0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.58)
     t_1
     (if (<= t 0.58) (+ 0.5 (* (* t t) (+ 1.0 (* t (+ t -2.0))))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.58d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (t + (-2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.58) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.58:
		tmp = t_1
	elif t <= 0.58:
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * Float64(t + -2.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.58)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = 0.5 + ((t * t) * (1.0 + (t * (t + -2.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.58], t$95$1, If[LessEqual[t, 0.58], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * N[(t + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.57999999999999996 or 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

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right) - \color{blue}{\frac{2}{9}} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} \]

      4. associate--r- N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right) \]

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right) \]

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}} \]

      12. unsub-neg N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + -1 \cdot \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right)}{\color{blue}{t}}\right)\right) \]

   5. Simplified 99.6%

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

   if -0.57999999999999996 < 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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + t \cdot \left(t - 2\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + t \cdot \left(t - 2\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + t \cdot \left(t - 2\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(t - 2\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(t - 2\right)}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(t + -2\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(-2 + \color{blue}{t}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(-2, \color{blue}{t}\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(-2 + t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.58:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot \left(t + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.6)
     t_1
     (if (<= t 0.44) (+ 0.5 (* (* t t) (+ 1.0 (* t -2.0)))) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.6d0)) then
        tmp = t_1
    else if (t <= 0.44d0) then
        tmp = 0.5d0 + ((t * t) * (1.0d0 + (t * (-2.0d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.6) {
		tmp = t_1;
	} else if (t <= 0.44) {
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.6:
		tmp = t_1
	elif t <= 0.44:
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = Float64(0.5 + Float64(Float64(t * t) * Float64(1.0 + Float64(t * -2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.6)
		tmp = t_1;
	elseif (t <= 0.44)
		tmp = 0.5 + ((t * t) * (1.0 + (t * -2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.6], t$95$1, If[LessEqual[t, 0.44], N[(0.5 + N[(N[(t * t), $MachinePrecision] * N[(1.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.440000000000000002 < 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

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right) - \color{blue}{\frac{2}{9}} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} \]

      4. associate--r- N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right) \]

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right) \]

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}} \]

      12. unsub-neg N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + -1 \cdot \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right)}{\color{blue}{t}}\right)\right) \]

   5. Simplified 99.6%

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

   if -0.599999999999999978 < 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 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({t}^{2} \cdot \left(1 + -2 \cdot t\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({t}^{2}\right), \color{blue}{\left(1 + -2 \cdot t\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(t \cdot t\right), \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\color{blue}{1} + -2 \cdot t\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(-2 \cdot t + \color{blue}{1}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\left(-2 \cdot t\right), \color{blue}{1}\right)\right)\right) \]

      7. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, t\right), 1\right)\right)\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{0.5 + \left(t \cdot t\right) \cdot \left(-2 \cdot t + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;0.5 + \left(t \cdot t\right) \cdot \left(1 + t \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1
         (+
          0.8333333333333334
          (/ (+ -0.2222222222222222 (/ 0.037037037037037035 t)) t))))
   (if (<= t -0.82) t_1 (if (<= t 0.235) (+ (* t t) 0.5) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + (((-0.2222222222222222d0) + (0.037037037037037035d0 / t)) / t)
    if (t <= (-0.82d0)) then
        tmp = t_1
    else if (t <= 0.235d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	double tmp;
	if (t <= -0.82) {
		tmp = t_1;
	} else if (t <= 0.235) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t)
	tmp = 0
	if t <= -0.82:
		tmp = t_1
	elif t <= 0.235:
		tmp = (t * t) + 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(Float64(-0.2222222222222222 + Float64(0.037037037037037035 / t)) / t))
	tmp = 0.0
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.235)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + ((-0.2222222222222222 + (0.037037037037037035 / t)) / t);
	tmp = 0.0;
	if (t <= -0.82)
		tmp = t_1;
	elseif (t <= 0.235)
		tmp = (t * t) + 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(N[(-0.2222222222222222 + N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.82], t$95$1, If[LessEqual[t, 0.235], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.819999999999999951 or 0.23499999999999999 < 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

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{5}{6}\right) - \color{blue}{\frac{2}{9}} \cdot \frac{1}{t} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\frac{1}{27}}{{t}^{2}} + \color{blue}{\left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}\right) + \color{blue}{\frac{\frac{1}{27}}{{t}^{2}}} \]

      4. associate--r- N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\left(\frac{2}{9} \cdot \frac{1}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9} \cdot 1}{t} - \frac{\color{blue}{\frac{1}{27}}}{{t}^{2}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{{t}^{2}}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27}}{t \cdot \color{blue}{t}}\right) \]

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27}}{t}}{\color{blue}{t}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{\frac{1}{27} \cdot 1}{t}}{t}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \frac{\frac{1}{27} \cdot \frac{1}{t}}{t}\right) \]

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{\color{blue}{t}} \]

      12. unsub-neg N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto \frac{5}{6} + -1 \cdot \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(-1 \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)}\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{-1 \cdot \left(\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}\right)}{\color{blue}{t}}\right)\right) \]

   5. Simplified 99.6%

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

   if -0.819999999999999951 < t < 0.23499999999999999

   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

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {t}^{2} + \color{blue}{\frac{1}{2}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({t}^{2}\right), \color{blue}{\frac{1}{2}}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot t\right), \frac{1}{2}\right) \]

      4. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{2}\right) \]

   5. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.82:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.235:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222 + \frac{0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.58:\\ \;\;\;\;t \cdot t + 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.8) t_1 (if (<= t 0.58) (+ (* t t) 0.5) t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.8d0)) then
        tmp = t_1
    else if (t <= 0.58d0) then
        tmp = (t * t) + 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.8) {
		tmp = t_1;
	} else if (t <= 0.58) {
		tmp = (t * t) + 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.8:
		tmp = t_1
	elif t <= 0.58:
		tmp = (t * t) + 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = Float64(Float64(t * t) + 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.8)
		tmp = t_1;
	elseif (t <= 0.58)
		tmp = (t * t) + 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.8], t$95$1, If[LessEqual[t, 0.58], N[(N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.80000000000000004 or 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

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 99.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   5. Simplified 99.2%

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

   if -0.80000000000000004 < 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

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {t}^{2} + \color{blue}{\frac{1}{2}} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left({t}^{2}\right), \color{blue}{\frac{1}{2}}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot t\right), \frac{1}{2}\right) \]

      4. *-lowering-*.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{2}\right) \]

   5. Simplified 99.6%

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

Alternative 10: 99.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.67:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 0.8333333333333334 (/ -0.2222222222222222 t))))
   (if (<= t -0.5) t_1 (if (<= t 0.67) 0.5 t_1))))

C

double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.67) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.8333333333333334d0 + ((-0.2222222222222222d0) / t)
    if (t <= (-0.5d0)) then
        tmp = t_1
    else if (t <= 0.67d0) then
        tmp = 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	double tmp;
	if (t <= -0.5) {
		tmp = t_1;
	} else if (t <= 0.67) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(t):
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t)
	tmp = 0
	if t <= -0.5:
		tmp = t_1
	elif t <= 0.67:
		tmp = 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(t)
	t_1 = Float64(0.8333333333333334 + Float64(-0.2222222222222222 / t))
	tmp = 0.0
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.67)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 0.8333333333333334 + (-0.2222222222222222 / t);
	tmp = 0.0;
	if (t <= -0.5)
		tmp = t_1;
	elseif (t <= 0.67)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.5], t$95$1, If[LessEqual[t, 0.67], 0.5, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.5 or 0.67000000000000004 < 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

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{5}{6} + \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \color{blue}{\left(\mathsf{neg}\left(\frac{2}{9} \cdot \frac{1}{t}\right)\right)}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9} \cdot 1}{t}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\mathsf{neg}\left(\frac{\frac{2}{9}}{t}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \left(\frac{\mathsf{neg}\left(\frac{2}{9}\right)}{\color{blue}{t}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{2}{9}\right)\right), \color{blue}{t}\right)\right) \]

      7. metadata-eval 99.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{5}{6}, \mathsf{/.f64}\left(\frac{-2}{9}, t\right)\right) \]

   5. Simplified 99.2%

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

   if -0.5 < t < 0.67000000000000004

   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

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

Alternative 11: 98.6% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		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 <= (-0.34d0)) 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 <= -0.34) {
		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 <= -0.34:
		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 <= -0.34)
		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 <= -0.34)
		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, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 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

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Simplified 98.4%

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

   if -0.340000000000000024 < 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

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

Alternative 12: 58.9% 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

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Simplified 55.7%

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

Reproduce

herbie shell --seed 2024185 
(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))))))))