Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 8.6s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 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 * t) / (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 * t) / (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 * t) / (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 * t) / (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(Float64(2.0 * t) / 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 * t) / (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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 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 * t) / (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 * t) / (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 * t) / (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 * t) / (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(Float64(2.0 * t) / 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 * t) / (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[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-1 - t}\\ t_2 := 4 \cdot \left(\frac{t}{t + 1} \cdot t\_1\right)\\ t_3 := 1 - t\_2\\ \mathbf{if}\;t \leq -0.7:\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{2 - t\_2}\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;\frac{t\_3}{2 - 4 \cdot \left(t \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (- -1.0 t)))
        (t_2 (* 4.0 (* (/ t (+ t 1.0)) t_1)))
        (t_3 (- 1.0 t_2)))
   (if (<= t -0.7)
     (/ (+ 5.0 (/ (- (/ (+ 12.0 (/ -16.0 t)) t) 8.0) t)) (- 2.0 t_2))
     (if (<= t 1.55)
       (/ t_3 (- 2.0 (* 4.0 (* t t_1))))
       (/ t_3 (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double t_1 = t / (-1.0 - t);
	double t_2 = 4.0 * ((t / (t + 1.0)) * t_1);
	double t_3 = 1.0 - t_2;
	double tmp;
	if (t <= -0.7) {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (2.0 - t_2);
	} else if (t <= 1.55) {
		tmp = t_3 / (2.0 - (4.0 * (t * t_1)));
	} else {
		tmp = t_3 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = t / ((-1.0d0) - t)
    t_2 = 4.0d0 * ((t / (t + 1.0d0)) * t_1)
    t_3 = 1.0d0 - t_2
    if (t <= (-0.7d0)) then
        tmp = (5.0d0 + ((((12.0d0 + ((-16.0d0) / t)) / t) - 8.0d0) / t)) / (2.0d0 - t_2)
    else if (t <= 1.55d0) then
        tmp = t_3 / (2.0d0 - (4.0d0 * (t * t_1)))
    else
        tmp = t_3 / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t / (-1.0 - t);
	double t_2 = 4.0 * ((t / (t + 1.0)) * t_1);
	double t_3 = 1.0 - t_2;
	double tmp;
	if (t <= -0.7) {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (2.0 - t_2);
	} else if (t <= 1.55) {
		tmp = t_3 / (2.0 - (4.0 * (t * t_1)));
	} else {
		tmp = t_3 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t / (-1.0 - t)
	t_2 = 4.0 * ((t / (t + 1.0)) * t_1)
	t_3 = 1.0 - t_2
	tmp = 0
	if t <= -0.7:
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (2.0 - t_2)
	elif t <= 1.55:
		tmp = t_3 / (2.0 - (4.0 * (t * t_1)))
	else:
		tmp = t_3 / (6.0 - ((8.0 + (-12.0 / t)) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t / Float64(-1.0 - t))
	t_2 = Float64(4.0 * Float64(Float64(t / Float64(t + 1.0)) * t_1))
	t_3 = Float64(1.0 - t_2)
	tmp = 0.0
	if (t <= -0.7)
		tmp = Float64(Float64(5.0 + Float64(Float64(Float64(Float64(12.0 + Float64(-16.0 / t)) / t) - 8.0) / t)) / Float64(2.0 - t_2));
	elseif (t <= 1.55)
		tmp = Float64(t_3 / Float64(2.0 - Float64(4.0 * Float64(t * t_1))));
	else
		tmp = Float64(t_3 / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t / (-1.0 - t);
	t_2 = 4.0 * ((t / (t + 1.0)) * t_1);
	t_3 = 1.0 - t_2;
	tmp = 0.0;
	if (t <= -0.7)
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (2.0 - t_2);
	elseif (t <= 1.55)
		tmp = t_3 / (2.0 - (4.0 * (t * t_1)));
	else
		tmp = t_3 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - t$95$2), $MachinePrecision]}, If[LessEqual[t, -0.7], N[(N[(5.0 + N[(N[(N[(N[(12.0 + N[(-16.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(2.0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55], N[(t$95$3 / N[(2.0 - N[(4.0 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.69999999999999996

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{\color{blue}{5 + -1 \cdot \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{5 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{\color{blue}{5 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      3. mul-1-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      4. unsub-neg 100.0%

         \[\leadsto \frac{5 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      6. associate-*r/ 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      8. distribute-neg-frac 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}}{2 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)} \]

   if -0.69999999999999996 < t < 1.55000000000000004

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.1%

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

   if 1.55000000000000004 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.7:\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{2 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{-1 - t}\\ t_2 := 1 - 4 \cdot \left(\frac{t}{t + 1} \cdot t\_1\right)\\ \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;\frac{t\_2}{2 - 4 \cdot \left(t \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (- -1.0 t))) (t_2 (- 1.0 (* 4.0 (* (/ t (+ t 1.0)) t_1)))))
   (if (<= t -0.52)
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
     (if (<= t 1.55)
       (/ t_2 (- 2.0 (* 4.0 (* t t_1))))
       (/ t_2 (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double t_1 = t / (-1.0 - t);
	double t_2 = 1.0 - (4.0 * ((t / (t + 1.0)) * t_1));
	double tmp;
	if (t <= -0.52) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 1.55) {
		tmp = t_2 / (2.0 - (4.0 * (t * t_1)));
	} else {
		tmp = t_2 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	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 = t / ((-1.0d0) - t)
    t_2 = 1.0d0 - (4.0d0 * ((t / (t + 1.0d0)) * t_1))
    if (t <= (-0.52d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 1.55d0) then
        tmp = t_2 / (2.0d0 - (4.0d0 * (t * t_1)))
    else
        tmp = t_2 / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t / (-1.0 - t);
	double t_2 = 1.0 - (4.0 * ((t / (t + 1.0)) * t_1));
	double tmp;
	if (t <= -0.52) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 1.55) {
		tmp = t_2 / (2.0 - (4.0 * (t * t_1)));
	} else {
		tmp = t_2 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t / (-1.0 - t)
	t_2 = 1.0 - (4.0 * ((t / (t + 1.0)) * t_1))
	tmp = 0
	if t <= -0.52:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 1.55:
		tmp = t_2 / (2.0 - (4.0 * (t * t_1)))
	else:
		tmp = t_2 / (6.0 - ((8.0 + (-12.0 / t)) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t / Float64(-1.0 - t))
	t_2 = Float64(1.0 - Float64(4.0 * Float64(Float64(t / Float64(t + 1.0)) * t_1)))
	tmp = 0.0
	if (t <= -0.52)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 1.55)
		tmp = Float64(t_2 / Float64(2.0 - Float64(4.0 * Float64(t * t_1))));
	else
		tmp = Float64(t_2 / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t / (-1.0 - t);
	t_2 = 1.0 - (4.0 * ((t / (t + 1.0)) * t_1));
	tmp = 0.0;
	if (t <= -0.52)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 1.55)
		tmp = t_2 / (2.0 - (4.0 * (t * t_1)));
	else
		tmp = t_2 / (6.0 - ((8.0 + (-12.0 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(4.0 * N[(N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.52], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55], N[(t$95$2 / N[(2.0 - N[(4.0 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.52000000000000002

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 98.7%

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

      5. metadata-eval 98.7%

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

      6. distribute-neg-frac 98.7%

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

      7. metadata-eval 98.7%

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

   10. Simplified 98.7%

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

   if -0.52000000000000002 < t < 1.55000000000000004

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.7%

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

   if 1.55000000000000004 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.52:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 1.55:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t + -1\right)\\ \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 1.4:\\ \;\;\;\;\frac{4 \cdot \left(t\_1 \cdot t\_1\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ t -1.0))))
   (if (<= t -0.48)
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
     (if (<= t 1.4)
       (/ (+ (* 4.0 (* t_1 t_1)) 1.0) 2.0)
       (/
        (- 1.0 (* 4.0 (* (/ t (+ t 1.0)) (/ t (- -1.0 t)))))
        (- 6.0 (/ (+ 8.0 (/ -12.0 t)) t)))))))

C

double code(double t) {
	double t_1 = t * (t + -1.0);
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 1.4) {
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 - (4.0 * ((t / (t + 1.0)) * (t / (-1.0 - t))))) / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (t + (-1.0d0))
    if (t <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 1.4d0) then
        tmp = ((4.0d0 * (t_1 * t_1)) + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 - (4.0d0 * ((t / (t + 1.0d0)) * (t / ((-1.0d0) - t))))) / (6.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (t + -1.0);
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 1.4) {
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 - (4.0 * ((t / (t + 1.0)) * (t / (-1.0 - t))))) / (6.0 - ((8.0 + (-12.0 / t)) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (t + -1.0)
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 1.4:
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0
	else:
		tmp = (1.0 - (4.0 * ((t / (t + 1.0)) * (t / (-1.0 - t))))) / (6.0 - ((8.0 + (-12.0 / t)) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(t + -1.0))
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 1.4)
		tmp = Float64(Float64(Float64(4.0 * Float64(t_1 * t_1)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(4.0 * Float64(Float64(t / Float64(t + 1.0)) * Float64(t / Float64(-1.0 - t))))) / Float64(6.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (t + -1.0);
	tmp = 0.0;
	if (t <= -0.48)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 1.4)
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	else
		tmp = (1.0 - (4.0 * ((t / (t + 1.0)) * (t / (-1.0 - t))))) / (6.0 - ((8.0 + (-12.0 / t)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4], N[(N[(N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(4.0 * N[(N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.47999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 98.7%

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

      5. metadata-eval 98.7%

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

      6. distribute-neg-frac 98.7%

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

      7. metadata-eval 98.7%

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

   10. Simplified 98.7%

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

   if -0.47999999999999998 < t < 1.3999999999999999

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.1%

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

   6. Taylor expanded in t around 0 98.1%

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

      1. neg-mul-1 98.1%

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

      2. sub-neg 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in t around 0 98.1%

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

       1. neg-mul-1 98.1%

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

       2. sub-neg 98.1%

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

   11. Simplified 98.1%

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

   if 1.3999999999999999 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 1.4:\\ \;\;\;\;\frac{4 \cdot \left(\left(t \cdot \left(t + -1\right)\right) \cdot \left(t \cdot \left(t + -1\right)\right)\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 4 \cdot \left(\frac{t}{t + 1} \cdot \frac{t}{-1 - t}\right)}{6 - \frac{8 + \frac{-12}{t}}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(t + -1\right)\\ \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\frac{4 \cdot \left(t\_1 \cdot t\_1\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ t -1.0))))
   (if (<= t -0.48)
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
     (if (<= t 0.8)
       (/ (+ (* 4.0 (* t_1 t_1)) 1.0) 2.0)
       (- 0.8333333333333334 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double t_1 = t * (t + -1.0);
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.8) {
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (t + (-1.0d0))
    if (t <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 0.8d0) then
        tmp = ((4.0d0 * (t_1 * t_1)) + 1.0d0) / 2.0d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (t + -1.0);
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.8) {
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (t + -1.0)
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 0.8:
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(t + -1.0))
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 0.8)
		tmp = Float64(Float64(Float64(4.0 * Float64(t_1 * t_1)) + 1.0) / 2.0);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (t + -1.0);
	tmp = 0.0;
	if (t <= -0.48)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 0.8)
		tmp = ((4.0 * (t_1 * t_1)) + 1.0) / 2.0;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.8], N[(N[(N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.47999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 98.7%

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

      5. metadata-eval 98.7%

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

      6. distribute-neg-frac 98.7%

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

      7. metadata-eval 98.7%

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

   10. Simplified 98.7%

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

   if -0.47999999999999998 < t < 0.80000000000000004

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.1%

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

   6. Taylor expanded in t around 0 98.1%

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

      1. neg-mul-1 98.1%

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

      2. sub-neg 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in t around 0 98.1%

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

       1. neg-mul-1 98.1%

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

       2. sub-neg 98.1%

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

   11. Simplified 98.1%

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

   if 0.80000000000000004 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;\frac{4 \cdot \left(\left(t \cdot \left(t + -1\right)\right) \cdot \left(t \cdot \left(t + -1\right)\right)\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.7:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;\frac{1 - 4 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.7)
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (if (<= t 0.45)
     (/ (- 1.0 (* 4.0 (* t (* t t)))) 2.0)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.7) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.45) {
		tmp = (1.0 - (4.0 * (t * (t * t)))) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.7d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 0.45d0) then
        tmp = (1.0d0 - (4.0d0 * (t * (t * t)))) / 2.0d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.7) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.45) {
		tmp = (1.0 - (4.0 * (t * (t * t)))) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.7:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 0.45:
		tmp = (1.0 - (4.0 * (t * (t * t)))) / 2.0
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.7)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 0.45)
		tmp = Float64(Float64(1.0 - Float64(4.0 * Float64(t * Float64(t * t)))) / 2.0);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.7)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 0.45)
		tmp = (1.0 - (4.0 * (t * (t * t)))) / 2.0;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.7], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.45], N[(N[(1.0 - N[(4.0 * N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.69999999999999996

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.8%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 99.8%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

   if -0.69999999999999996 < t < 0.450000000000000011

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 97.5%

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

   6. Taylor expanded in t around 0 97.5%

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

      1. neg-mul-1 97.5%

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

      2. sub-neg 97.5%

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

   8. Simplified 97.5%

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

   9. Taylor expanded in t around 0 97.5%

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

   10. Taylor expanded in t around inf 97.5%

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

       1. neg-mul-1 97.5%

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

   12. Simplified 97.5%

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

   if 0.450000000000000011 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.7:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.45:\\ \;\;\;\;\frac{1 - 4 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.48:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.62)
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (if (<= t 0.48)
     (/ (+ (* 4.0 (* t t)) 1.0) 2.0)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.62) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.48) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.62d0)) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else if (t <= 0.48d0) then
        tmp = ((4.0d0 * (t * t)) + 1.0d0) / 2.0d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.62) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else if (t <= 0.48) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.62:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	elif t <= 0.48:
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.62)
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	elseif (t <= 0.48)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * t)) + 1.0) / 2.0);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.62)
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	elseif (t <= 0.48)
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.62], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.48], N[(N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.619999999999999996

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 + -1 \cdot \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 - 12 \cdot \frac{1}{t}}{t}\right)}} \]

      2. unsub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - 12 \cdot \frac{1}{t}}{t}}} \]

      3. sub-neg 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 + \left(-12 \cdot \frac{1}{t}\right)}}{t}} \]

      4. associate-*r/ 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\color{blue}{\frac{12 \cdot 1}{t}}\right)}{t}} \]

      5. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \left(-\frac{\color{blue}{12}}{t}\right)}{t}} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\frac{-12}{t}}}{t}} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \frac{\color{blue}{-12}}{t}}{t}} \]

   7. Simplified 98.5%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. associate-*r/ 98.7%

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

      5. metadata-eval 98.7%

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

      6. distribute-neg-frac 98.7%

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

      7. metadata-eval 98.7%

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

   10. Simplified 98.7%

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

   if -0.619999999999999996 < t < 0.47999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.1%

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

   6. Taylor expanded in t around 0 98.1%

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

      1. neg-mul-1 98.1%

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

      2. sub-neg 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in t around 0 98.1%

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

   10. Taylor expanded in t around 0 98.1%

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

   if 0.47999999999999998 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{elif}\;t \leq 0.48:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 0.8333333333333334 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.8333333333333334d0
end function

Java

public static double code(double t) {
	return 0.8333333333333334;
}

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

function tmp = code(t)
	tmp = 0.8333333333333334;
end

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around inf 59.2%

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

6. Taylor expanded in t around inf 60.3%

   \[\leadsto \color{blue}{0.8333333333333334} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024147 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))