Kahan p13 Example 1

Percentage Accurate: 100.0% → 100.0%

Time: 22.3s

Alternatives: 10

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: 100.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{1 + t}\\ \mathbf{if}\;t \leq -1.32 \lor \neg \left(t \leq 1.65\right):\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{4 \cdot \left(t\_1 \cdot t\_1\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ t (+ 1.0 t))))
   (if (or (<= t -1.32) (not (<= t 1.65)))
     (/
      (+ 5.0 (/ (- (/ (+ 12.0 (/ -16.0 t)) t) 8.0) t))
      (+ (* 4.0 (* t_1 t_1)) 2.0))
     (/
      (+ 1.0 (* 4.0 (* t (* t (- 1.0 (* t (- 1.0 t)))))))
      (+ 2.0 (* 4.0 (* t t)))))))

C

double code(double t) {
	double t_1 = t / (1.0 + t);
	double tmp;
	if ((t <= -1.32) || !(t <= 1.65)) {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / ((4.0 * (t_1 * t_1)) + 2.0);
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (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 / (1.0d0 + t)
    if ((t <= (-1.32d0)) .or. (.not. (t <= 1.65d0))) then
        tmp = (5.0d0 + ((((12.0d0 + ((-16.0d0) / t)) / t) - 8.0d0) / t)) / ((4.0d0 * (t_1 * t_1)) + 2.0d0)
    else
        tmp = (1.0d0 + (4.0d0 * (t * (t * (1.0d0 - (t * (1.0d0 - t))))))) / (2.0d0 + (4.0d0 * (t * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t / (1.0 + t);
	double tmp;
	if ((t <= -1.32) || !(t <= 1.65)) {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / ((4.0 * (t_1 * t_1)) + 2.0);
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t / (1.0 + t)
	tmp = 0
	if (t <= -1.32) or not (t <= 1.65):
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / ((4.0 * (t_1 * t_1)) + 2.0)
	else:
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)))
	return tmp

Julia

function code(t)
	t_1 = Float64(t / Float64(1.0 + t))
	tmp = 0.0
	if ((t <= -1.32) || !(t <= 1.65))
		tmp = Float64(Float64(5.0 + Float64(Float64(Float64(Float64(12.0 + Float64(-16.0 / t)) / t) - 8.0) / t)) / Float64(Float64(4.0 * Float64(t_1 * t_1)) + 2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(t * Float64(t * Float64(1.0 - Float64(t * Float64(1.0 - t))))))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t / (1.0 + t);
	tmp = 0.0;
	if ((t <= -1.32) || ~((t <= 1.65)))
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / ((4.0 * (t_1 * t_1)) + 2.0);
	else
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.32], N[Not[LessEqual[t, 1.65]], $MachinePrecision]], 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[(N[(4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(4.0 * N[(t * N[(t * N[(1.0 - N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32000000000000006 or 1.6499999999999999 < 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 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. unsub-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. associate-*r/ 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -1.32000000000000006 < t < 1.6499999999999999

   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 98.9%

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

   6. Taylor expanded in t around 0 98.9%

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

   7. Taylor expanded in t around 0 98.9%

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

   8. Taylor expanded in t around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \lor \neg \left(t \leq 1.65\right):\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}\\ \mathbf{if}\;t \leq -0.54 \lor \neg \left(t \leq 1.55\right):\\ \;\;\;\;\frac{5 + t\_1}{2 + \left(4 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (- (/ (+ 12.0 (/ -16.0 t)) t) 8.0) t)))
   (if (or (<= t -0.54) (not (<= t 1.55)))
     (/ (+ 5.0 t_1) (+ 2.0 (+ 4.0 t_1)))
     (/
      (+ 1.0 (* 4.0 (* t (* t (- 1.0 (* t (- 1.0 t)))))))
      (+ 2.0 (* 4.0 (* t t)))))))

C

double code(double t) {
	double t_1 = (((12.0 + (-16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.54) || !(t <= 1.55)) {
		tmp = (5.0 + t_1) / (2.0 + (4.0 + t_1));
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((12.0d0 + ((-16.0d0) / t)) / t) - 8.0d0) / t
    if ((t <= (-0.54d0)) .or. (.not. (t <= 1.55d0))) then
        tmp = (5.0d0 + t_1) / (2.0d0 + (4.0d0 + t_1))
    else
        tmp = (1.0d0 + (4.0d0 * (t * (t * (1.0d0 - (t * (1.0d0 - t))))))) / (2.0d0 + (4.0d0 * (t * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (((12.0 + (-16.0 / t)) / t) - 8.0) / t;
	double tmp;
	if ((t <= -0.54) || !(t <= 1.55)) {
		tmp = (5.0 + t_1) / (2.0 + (4.0 + t_1));
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Python

def code(t):
	t_1 = (((12.0 + (-16.0 / t)) / t) - 8.0) / t
	tmp = 0
	if (t <= -0.54) or not (t <= 1.55):
		tmp = (5.0 + t_1) / (2.0 + (4.0 + t_1))
	else:
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)))
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(Float64(12.0 + Float64(-16.0 / t)) / t) - 8.0) / t)
	tmp = 0.0
	if ((t <= -0.54) || !(t <= 1.55))
		tmp = Float64(Float64(5.0 + t_1) / Float64(2.0 + Float64(4.0 + t_1)));
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(t * Float64(t * Float64(1.0 - Float64(t * Float64(1.0 - t))))))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (((12.0 + (-16.0 / t)) / t) - 8.0) / t;
	tmp = 0.0;
	if ((t <= -0.54) || ~((t <= 1.55)))
		tmp = (5.0 + t_1) / (2.0 + (4.0 + t_1));
	else
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(N[(12.0 + N[(-16.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[Or[LessEqual[t, -0.54], N[Not[LessEqual[t, 1.55]], $MachinePrecision]], N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(2.0 + N[(4.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(4.0 * N[(t * N[(t * N[(1.0 - N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.54000000000000004 or 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 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. unsub-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. associate-*r/ 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. mul-1-neg 99.6%

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

      4. unsub-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. associate-*r/ 99.6%

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

      7. metadata-eval 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

   if -0.54000000000000004 < 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 98.9%

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

   6. Taylor expanded in t around 0 98.9%

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

   7. Taylor expanded in t around 0 98.9%

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

   8. Taylor expanded in t around 0 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.54 \lor \neg \left(t \leq 1.55\right):\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{2 + \left(4 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.65 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.65) (not (<= t 0.75)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (/
    (+ 1.0 (* 4.0 (* t (* t (- 1.0 (* t (- 1.0 t)))))))
    (+ 2.0 (* 4.0 (* t t))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.65) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.65d0)) .or. (.not. (t <= 0.75d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = (1.0d0 + (4.0d0 * (t * (t * (1.0d0 - (t * (1.0d0 - t))))))) / (2.0d0 + (4.0d0 * (t * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.65) || !(t <= 0.75)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.65) or not (t <= 0.75):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.65) || !(t <= 0.75))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(t * Float64(t * Float64(1.0 - Float64(t * Float64(1.0 - t))))))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.65) || ~((t <= 0.75)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = (1.0 + (4.0 * (t * (t * (1.0 - (t * (1.0 - t))))))) / (2.0 + (4.0 * (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.65], N[Not[LessEqual[t, 0.75]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(4.0 * N[(t * N[(t * N[(1.0 - N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.650000000000000022 or 0.75 < 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 99.4%

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

      1. mul-1-neg 99.4%

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

      2. unsub-neg 99.4%

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

      3. mul-1-neg 99.4%

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

      4. unsub-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. associate-*r/ 99.4%

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

      7. metadata-eval 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in t around -inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. associate-*r/ 99.0%

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

      5. metadata-eval 99.0%

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

      6. distribute-neg-frac 99.0%

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

      7. metadata-eval 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in t around inf 99.3%

       \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   12. Step-by-step derivation

       1. sub-neg 99.3%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)} \]

       2. +-commutative 99.3%

          \[\leadsto \color{blue}{\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       3. associate-+r+ 99.3%

          \[\leadsto \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + 0.8333333333333334\right) + \frac{0.037037037037037035}{{t}^{2}}} \]

       4. +-commutative 99.3%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       5. sub-neg 99.3%

          \[\leadsto \color{blue}{\left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       6. associate--r- 99.3%

          \[\leadsto \color{blue}{0.8333333333333334 - \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       7. associate-*r/ 99.3%

          \[\leadsto 0.8333333333333334 - \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       8. metadata-eval 99.3%

          \[\leadsto 0.8333333333333334 - \left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       9. unpow2 99.3%

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

       10. associate-/r* 99.3%

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

       11. metadata-eval 99.3%

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

       12. associate-*r/ 99.3%

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

       13. div-sub 99.3%

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

       14. sub-neg 99.3%

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

       15. associate-*r/ 99.3%

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

       16. metadata-eval 99.3%

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

       17. distribute-neg-frac 99.3%

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

       18. metadata-eval 99.3%

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

   13. Simplified 99.3%

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

   if -0.650000000000000022 < t < 0.75

   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 98.9%

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

   6. Taylor expanded in t around 0 98.9%

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

   7. Taylor expanded in t around 0 98.9%

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

   8. Taylor expanded in t around 0 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.65 \lor \neg \left(t \leq 0.75\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \left(t \cdot \left(1 - t \cdot \left(1 - t\right)\right)\right)\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \frac{t}{1 + t}\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.6) (not (<= t 0.56)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (/ (+ 1.0 (* 4.0 (* t (/ t (+ 1.0 t))))) (+ 2.0 (* 4.0 (* t t))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.6) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + (4.0 * (t * (t / (1.0 + t))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.6d0)) .or. (.not. (t <= 0.56d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = (1.0d0 + (4.0d0 * (t * (t / (1.0d0 + t))))) / (2.0d0 + (4.0d0 * (t * t)))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.6) || !(t <= 0.56)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + (4.0 * (t * (t / (1.0 + t))))) / (2.0 + (4.0 * (t * t)));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.6) or not (t <= 0.56):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = (1.0 + (4.0 * (t * (t / (1.0 + t))))) / (2.0 + (4.0 * (t * t)))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.6) || !(t <= 0.56))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(Float64(1.0 + Float64(4.0 * Float64(t * Float64(t / Float64(1.0 + t))))) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.6) || ~((t <= 0.56)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = (1.0 + (4.0 * (t * (t / (1.0 + t))))) / (2.0 + (4.0 * (t * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.6], N[Not[LessEqual[t, 0.56]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(4.0 * N[(t * N[(t / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.599999999999999978 or 0.56000000000000005 < 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 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. mul-1-neg 98.6%

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

      4. unsub-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. associate-*r/ 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. associate-*r/ 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-neg-frac 98.2%

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

      7. metadata-eval 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in t around inf 98.7%

       \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   12. Step-by-step derivation

       1. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)} \]

       2. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       3. associate-+r+ 98.7%

          \[\leadsto \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + 0.8333333333333334\right) + \frac{0.037037037037037035}{{t}^{2}}} \]

       4. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       5. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       6. associate--r- 98.7%

          \[\leadsto \color{blue}{0.8333333333333334 - \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       7. associate-*r/ 98.7%

          \[\leadsto 0.8333333333333334 - \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       8. metadata-eval 98.7%

          \[\leadsto 0.8333333333333334 - \left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       9. unpow2 98.7%

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

       10. associate-/r* 98.7%

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

       11. metadata-eval 98.7%

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

       12. associate-*r/ 98.7%

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

       13. div-sub 98.7%

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

       14. sub-neg 98.7%

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

       15. associate-*r/ 98.7%

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

       16. metadata-eval 98.7%

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

       17. distribute-neg-frac 98.7%

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

       18. metadata-eval 98.7%

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

   13. Simplified 98.7%

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

   if -0.599999999999999978 < t < 0.56000000000000005

   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 99.5%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in t around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.56\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot \frac{t}{1 + t}\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 4 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -0.64 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_1}{2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* 4.0 (* t t))))
   (if (or (<= t -0.64) (not (<= t 0.24)))
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
     (/ (+ 1.0 t_1) (+ 2.0 t_1)))))

C

double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if ((t <= -0.64) || !(t <= 0.24)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 4.0d0 * (t * t)
    if ((t <= (-0.64d0)) .or. (.not. (t <= 0.24d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = (1.0d0 + t_1) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = 4.0 * (t * t);
	double tmp;
	if ((t <= -0.64) || !(t <= 0.24)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = (1.0 + t_1) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t):
	t_1 = 4.0 * (t * t)
	tmp = 0
	if (t <= -0.64) or not (t <= 0.24):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = (1.0 + t_1) / (2.0 + t_1)
	return tmp

Julia

function code(t)
	t_1 = Float64(4.0 * Float64(t * t))
	tmp = 0.0
	if ((t <= -0.64) || !(t <= 0.24))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(Float64(1.0 + t_1) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = 4.0 * (t * t);
	tmp = 0.0;
	if ((t <= -0.64) || ~((t <= 0.24)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = (1.0 + t_1) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -0.64], N[Not[LessEqual[t, 0.24]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.640000000000000013 or 0.23999999999999999 < 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 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. mul-1-neg 98.6%

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

      4. unsub-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. associate-*r/ 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. associate-*r/ 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-neg-frac 98.2%

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

      7. metadata-eval 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in t around inf 98.7%

       \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   12. Step-by-step derivation

       1. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)} \]

       2. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       3. associate-+r+ 98.7%

          \[\leadsto \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + 0.8333333333333334\right) + \frac{0.037037037037037035}{{t}^{2}}} \]

       4. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       5. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       6. associate--r- 98.7%

          \[\leadsto \color{blue}{0.8333333333333334 - \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       7. associate-*r/ 98.7%

          \[\leadsto 0.8333333333333334 - \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       8. metadata-eval 98.7%

          \[\leadsto 0.8333333333333334 - \left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       9. unpow2 98.7%

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

       10. associate-/r* 98.7%

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

       11. metadata-eval 98.7%

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

       12. associate-*r/ 98.7%

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

       13. div-sub 98.7%

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

       14. sub-neg 98.7%

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

       15. associate-*r/ 98.7%

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

       16. metadata-eval 98.7%

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

       17. distribute-neg-frac 98.7%

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

       18. metadata-eval 98.7%

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

   13. Simplified 98.7%

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

   if -0.640000000000000013 < t < 0.23999999999999999

   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 99.5%

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

   6. Taylor expanded in t around 0 99.5%

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

   7. Taylor expanded in t around 0 99.5%

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

   8. Taylor expanded in t around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.64 \lor \neg \left(t \leq 0.24\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + 4 \cdot \left(t \cdot t\right)}{2 + 4 \cdot \left(t \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.52000000000000002 or 0.23000000000000001 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 98.6%

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

      1. mul-1-neg 98.6%

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

      2. unsub-neg 98.6%

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

      3. mul-1-neg 98.6%

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

      4. unsub-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. associate-*r/ 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-neg-frac 98.6%

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

      9. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in t around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. associate-*r/ 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-neg-frac 98.2%

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

      7. metadata-eval 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in t around inf 98.7%

       \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) - 0.2222222222222222 \cdot \frac{1}{t}} \]
   12. Step-by-step derivation

       1. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)} \]

       2. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + \left(0.8333333333333334 + \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       3. associate-+r+ 98.7%

          \[\leadsto \color{blue}{\left(\left(-0.2222222222222222 \cdot \frac{1}{t}\right) + 0.8333333333333334\right) + \frac{0.037037037037037035}{{t}^{2}}} \]

       4. +-commutative 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       5. sub-neg 98.7%

          \[\leadsto \color{blue}{\left(0.8333333333333334 - 0.2222222222222222 \cdot \frac{1}{t}\right)} + \frac{0.037037037037037035}{{t}^{2}} \]

       6. associate--r- 98.7%

          \[\leadsto \color{blue}{0.8333333333333334 - \left(0.2222222222222222 \cdot \frac{1}{t} - \frac{0.037037037037037035}{{t}^{2}}\right)} \]

       7. associate-*r/ 98.7%

          \[\leadsto 0.8333333333333334 - \left(\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       8. metadata-eval 98.7%

          \[\leadsto 0.8333333333333334 - \left(\frac{\color{blue}{0.2222222222222222}}{t} - \frac{0.037037037037037035}{{t}^{2}}\right) \]

       9. unpow2 98.7%

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

       10. associate-/r* 98.7%

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

       11. metadata-eval 98.7%

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

       12. associate-*r/ 98.7%

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

       13. div-sub 98.7%

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

       14. sub-neg 98.7%

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

       15. associate-*r/ 98.7%

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

       16. metadata-eval 98.7%

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

       17. distribute-neg-frac 98.7%

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

       18. metadata-eval 98.7%

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

   13. Simplified 98.7%

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

   if -0.52000000000000002 < t < 0.23000000000000001

   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 99.5%

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

   6. Taylor expanded in t around 0 99.4%

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

4. Final simplification 99.1%

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

Alternative 8: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < 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 97.2%

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

      1. associate-*r/ 97.2%

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

      2. metadata-eval 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in t around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. metadata-eval 97.8%

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

   10. Simplified 97.8%

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

   if -0.48999999999999999 < t < 0.680000000000000049

   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 99.5%

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

   6. Taylor expanded in t around 0 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 96.0%

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

   6. Taylor expanded in t around inf 96.3%

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

   if -0.330000000000000016 < t < 1

   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 98.9%

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

   6. Taylor expanded in t around 0 98.9%

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

4. Final simplification 97.6%

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

Alternative 10: 59.3% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

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

Wolfram

code[t_] := 0.5

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
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 52.8%

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

6. Taylor expanded in t around 0 60.5%

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

7. Final simplification 60.5%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))