Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 22.2s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double t) {
	double t_1 = t / (t + 1.0);
	double t_2 = 4.0 * (t_1 * t_1);
	return (t_2 + 1.0) / (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 / (t + 1.0d0)
    t_2 = 4.0d0 * (t_1 * t_1)
    code = (t_2 + 1.0d0) / (t_2 + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.38:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \left(t \cdot \left(1 - t\right)\right)\right) + 1}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{6 - \frac{\frac{\frac{16}{t} - 12}{t} - -8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.38)
   0.8333333333333334
   (if (<= t 0.86)
     (/
      (+ (* 4.0 (* t (* t (- 1.0 t)))) 1.0)
      (- 2.0 (* 4.0 (* t (/ t (- -1.0 t))))))
     (/
      (+ 5.0 (/ (- (/ (+ 12.0 (/ -16.0 t)) t) 8.0) t))
      (- 6.0 (/ (- (/ (- (/ 16.0 t) 12.0) t) -8.0) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.38) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.86) {
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (6.0 - (((((16.0 / t) - 12.0) / t) - -8.0) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.38d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.86d0) then
        tmp = ((4.0d0 * (t * (t * (1.0d0 - t)))) + 1.0d0) / (2.0d0 - (4.0d0 * (t * (t / ((-1.0d0) - t)))))
    else
        tmp = (5.0d0 + ((((12.0d0 + ((-16.0d0) / t)) / t) - 8.0d0) / t)) / (6.0d0 - (((((16.0d0 / t) - 12.0d0) / t) - (-8.0d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.38) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.86) {
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (6.0 - (((((16.0 / t) - 12.0) / t) - -8.0) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.38:
		tmp = 0.8333333333333334
	elif t <= 0.86:
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))))
	else:
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (6.0 - (((((16.0 / t) - 12.0) / t) - -8.0) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.38)
		tmp = 0.8333333333333334;
	elseif (t <= 0.86)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * Float64(t * Float64(1.0 - t)))) + 1.0) / Float64(2.0 - Float64(4.0 * Float64(t * Float64(t / Float64(-1.0 - t))))));
	else
		tmp = Float64(Float64(5.0 + Float64(Float64(Float64(Float64(12.0 + Float64(-16.0 / t)) / t) - 8.0) / t)) / Float64(6.0 - Float64(Float64(Float64(Float64(Float64(16.0 / t) - 12.0) / t) - -8.0) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.38)
		tmp = 0.8333333333333334;
	elseif (t <= 0.86)
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	else
		tmp = (5.0 + ((((12.0 + (-16.0 / t)) / t) - 8.0) / t)) / (6.0 - (((((16.0 / t) - 12.0) / t) - -8.0) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.38], 0.8333333333333334, If[LessEqual[t, 0.86], N[(N[(N[(4.0 * N[(t * N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 - N[(4.0 * N[(t * N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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[(6.0 - N[(N[(N[(N[(N[(16.0 / t), $MachinePrecision] - 12.0), $MachinePrecision] / t), $MachinePrecision] - -8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.38

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   6. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

   if -0.38 < t < 0.859999999999999987

   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%

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

      1. neg-mul-1 99.5%

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

      2. sub-neg 99.5%

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

   9. Simplified 99.5%

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

   if 0.859999999999999987 < 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 98.7%

      \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) - \left(8 \cdot \frac{1}{t} + 16 \cdot \frac{1}{{t}^{3}}\right)}} \]

   10. Simplified 98.7%

       \[\leadsto \frac{5 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}{\color{blue}{6 + \frac{-8 + \frac{12 - \frac{16}{t}}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.38:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \left(t \cdot \left(1 - t\right)\right)\right) + 1}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + \frac{\frac{12 + \frac{-16}{t}}{t} - 8}{t}}{6 - \frac{\frac{\frac{16}{t} - 12}{t} - -8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.38:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.82:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \left(t \cdot \left(1 - t\right)\right)\right) + 1}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.38)
   0.8333333333333334
   (if (<= t 0.82)
     (/
      (+ (* 4.0 (* t (* t (- 1.0 t)))) 1.0)
      (- 2.0 (* 4.0 (* t (/ t (- -1.0 t))))))
     (/ (- 5.0 (/ (+ 8.0 (/ -12.0 t)) t)) (+ 6.0 (/ (- (/ 12.0 t) 8.0) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.38) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.82) {
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.38d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.82d0) then
        tmp = ((4.0d0 * (t * (t * (1.0d0 - t)))) + 1.0d0) / (2.0d0 - (4.0d0 * (t * (t / ((-1.0d0) - t)))))
    else
        tmp = (5.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t)) / (6.0d0 + (((12.0d0 / t) - 8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.38) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.82) {
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	} else {
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.38:
		tmp = 0.8333333333333334
	elif t <= 0.82:
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))))
	else:
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.38)
		tmp = 0.8333333333333334;
	elseif (t <= 0.82)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * Float64(t * Float64(1.0 - t)))) + 1.0) / Float64(2.0 - Float64(4.0 * Float64(t * Float64(t / Float64(-1.0 - t))))));
	else
		tmp = Float64(Float64(5.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t)) / Float64(6.0 + Float64(Float64(Float64(12.0 / t) - 8.0) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.38)
		tmp = 0.8333333333333334;
	elseif (t <= 0.82)
		tmp = ((4.0 * (t * (t * (1.0 - t)))) + 1.0) / (2.0 - (4.0 * (t * (t / (-1.0 - t)))));
	else
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.38], 0.8333333333333334, If[LessEqual[t, 0.82], N[(N[(N[(4.0 * N[(t * N[(t * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 - N[(4.0 * N[(t * N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(N[(N[(12.0 / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.38

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   6. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

   if -0.38 < t < 0.819999999999999951

   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%

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

      1. neg-mul-1 99.5%

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

      2. sub-neg 99.5%

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

   9. Simplified 99.5%

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

   if 0.819999999999999951 < 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 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. associate-*r/ 98.3%

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

      5. metadata-eval 98.3%

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

      6. distribute-neg-frac 98.3%

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

      7. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in t around inf 98.4%

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

      1. associate--l+ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{{t}^{2}} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      3. metadata-eval 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{{t}^{2}} - \frac{\color{blue}{8}}{t}\right)} \]

      4. unpow2 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{\color{blue}{t \cdot t}} - \frac{8}{t}\right)} \]

      5. associate-/r* 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\color{blue}{\frac{\frac{12}{t}}{t}} - \frac{8}{t}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t} - \frac{8}{t}\right)} \]

      7. associate-*r/ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{\color{blue}{12 \cdot \frac{1}{t}}}{t} - \frac{8}{t}\right)} \]

      8. div-sub 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \color{blue}{\frac{12 \cdot \frac{1}{t} - 8}{t}}} \]

      9. sub-neg 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{12 \cdot \frac{1}{t} + \left(-8\right)}}{t}} \]

      10. associate-*r/ 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{\frac{12 \cdot 1}{t}} + \left(-8\right)}{t}} \]

      11. metadata-eval 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{\color{blue}{12}}{t} + \left(-8\right)}{t}} \]

      12. metadata-eval 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{\color{blue}{--12}}{t} + \left(-8\right)}{t}} \]

      13. distribute-neg-frac 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{\left(-\frac{-12}{t}\right)} + \left(-8\right)}{t}} \]

      14. distribute-neg-in 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{-\left(\frac{-12}{t} + 8\right)}}{t}} \]

      15. +-commutative 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{-\color{blue}{\left(8 + \frac{-12}{t}\right)}}{t}} \]

      16. distribute-neg-frac 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \color{blue}{\left(-\frac{8 + \frac{-12}{t}}{t}\right)}} \]

      17. sub-neg 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   10. Simplified 98.4%

       \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 - \frac{8 - \frac{12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.38:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.82:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \left(t \cdot \left(1 - t\right)\right)\right) + 1}{2 - 4 \cdot \left(t \cdot \frac{t}{-1 - t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.85:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \frac{t}{t + 1}\right) + 1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.43)
   0.8333333333333334
   (if (<= t 1.85)
     (/ (+ (* 4.0 (* t (/ t (+ t 1.0)))) 1.0) (+ 2.0 (* 4.0 (* t t))))
     (/ (- 5.0 (/ 8.0 t)) (- 6.0 (/ 8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.43) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.85) {
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.43d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.85d0) then
        tmp = ((4.0d0 * (t * (t / (t + 1.0d0)))) + 1.0d0) / (2.0d0 + (4.0d0 * (t * t)))
    else
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 - (8.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.43) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.85) {
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.43:
		tmp = 0.8333333333333334
	elif t <= 1.85:
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)))
	else:
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.43)
		tmp = 0.8333333333333334;
	elseif (t <= 1.85)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * Float64(t / Float64(t + 1.0)))) + 1.0) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	else
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 - Float64(8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.43)
		tmp = 0.8333333333333334;
	elseif (t <= 1.85)
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	else
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.43], 0.8333333333333334, If[LessEqual[t, 1.85], N[(N[(N[(4.0 * N[(t * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.429999999999999993

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   6. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

   if -0.429999999999999993 < t < 1.8500000000000001

   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%

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

   if 1.8500000000000001 < 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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in t around inf 97.6%

      \[\leadsto \frac{5 - \frac{8}{t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.6%

         \[\leadsto \frac{5 - \frac{8}{t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 97.6%

         \[\leadsto \frac{5 - \frac{8}{t}}{6 - \frac{\color{blue}{8}}{t}} \]

   10. Simplified 97.6%

       \[\leadsto \frac{5 - \frac{8}{t}}{\color{blue}{6 - \frac{8}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.85:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \frac{t}{t + 1}\right) + 1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.1:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \frac{t}{t + 1}\right) + 1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.43)
   0.8333333333333334
   (if (<= t 1.1)
     (/ (+ (* 4.0 (* t (/ t (+ t 1.0)))) 1.0) (+ 2.0 (* 4.0 (* t t))))
     (/ (- 5.0 (/ (+ 8.0 (/ -12.0 t)) t)) (+ 6.0 (/ (- (/ 12.0 t) 8.0) t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.43) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.1) {
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.43d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.1d0) then
        tmp = ((4.0d0 * (t * (t / (t + 1.0d0)))) + 1.0d0) / (2.0d0 + (4.0d0 * (t * t)))
    else
        tmp = (5.0d0 - ((8.0d0 + ((-12.0d0) / t)) / t)) / (6.0d0 + (((12.0d0 / t) - 8.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.43) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.1) {
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	} else {
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.43:
		tmp = 0.8333333333333334
	elif t <= 1.1:
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)))
	else:
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.43)
		tmp = 0.8333333333333334;
	elseif (t <= 1.1)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * Float64(t / Float64(t + 1.0)))) + 1.0) / Float64(2.0 + Float64(4.0 * Float64(t * t))));
	else
		tmp = Float64(Float64(5.0 - Float64(Float64(8.0 + Float64(-12.0 / t)) / t)) / Float64(6.0 + Float64(Float64(Float64(12.0 / t) - 8.0) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.43)
		tmp = 0.8333333333333334;
	elseif (t <= 1.1)
		tmp = ((4.0 * (t * (t / (t + 1.0)))) + 1.0) / (2.0 + (4.0 * (t * t)));
	else
		tmp = (5.0 - ((8.0 + (-12.0 / t)) / t)) / (6.0 + (((12.0 / t) - 8.0) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.43], 0.8333333333333334, If[LessEqual[t, 1.1], N[(N[(N[(4.0 * N[(t * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 - N[(N[(8.0 + N[(-12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(N[(N[(12.0 / t), $MachinePrecision] - 8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.429999999999999993

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   6. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

   if -0.429999999999999993 < t < 1.1000000000000001

   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%

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

   if 1.1000000000000001 < 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 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. associate-*r/ 98.3%

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

      5. metadata-eval 98.3%

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

      6. distribute-neg-frac 98.3%

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

      7. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in t around inf 98.4%

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

      1. associate--l+ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 + \left(\frac{12}{{t}^{2}} - 8 \cdot \frac{1}{t}\right)}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{{t}^{2}} - \color{blue}{\frac{8 \cdot 1}{t}}\right)} \]

      3. metadata-eval 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{{t}^{2}} - \frac{\color{blue}{8}}{t}\right)} \]

      4. unpow2 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{12}{\color{blue}{t \cdot t}} - \frac{8}{t}\right)} \]

      5. associate-/r* 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\color{blue}{\frac{\frac{12}{t}}{t}} - \frac{8}{t}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{\frac{\color{blue}{12 \cdot 1}}{t}}{t} - \frac{8}{t}\right)} \]

      7. associate-*r/ 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \left(\frac{\color{blue}{12 \cdot \frac{1}{t}}}{t} - \frac{8}{t}\right)} \]

      8. div-sub 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \color{blue}{\frac{12 \cdot \frac{1}{t} - 8}{t}}} \]

      9. sub-neg 98.4%

         \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{12 \cdot \frac{1}{t} + \left(-8\right)}}{t}} \]

      10. associate-*r/ 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{\frac{12 \cdot 1}{t}} + \left(-8\right)}{t}} \]

      11. metadata-eval 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{\color{blue}{12}}{t} + \left(-8\right)}{t}} \]

      12. metadata-eval 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{\color{blue}{--12}}{t} + \left(-8\right)}{t}} \]

      13. distribute-neg-frac 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{\left(-\frac{-12}{t}\right)} + \left(-8\right)}{t}} \]

      14. distribute-neg-in 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\color{blue}{-\left(\frac{-12}{t} + 8\right)}}{t}} \]

      15. +-commutative 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{-\color{blue}{\left(8 + \frac{-12}{t}\right)}}{t}} \]

      16. distribute-neg-frac 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \color{blue}{\left(-\frac{8 + \frac{-12}{t}}{t}\right)}} \]

      17. sub-neg 98.4%

          \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 - \frac{8 + \frac{-12}{t}}{t}}} \]

   10. Simplified 98.4%

       \[\leadsto \frac{5 - \frac{8 + \frac{-12}{t}}{t}}{\color{blue}{6 - \frac{8 - \frac{12}{t}}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.43:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1.1:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot \frac{t}{t + 1}\right) + 1}{2 + 4 \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8 + \frac{-12}{t}}{t}}{6 + \frac{\frac{12}{t} - 8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34)
   0.8333333333333334
   (if (<= t 2.0) 0.5 (/ (- 5.0 (/ 8.0 t)) (- 6.0 (/ 8.0 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (5.0d0 - (8.0d0 / t)) / (6.0d0 - (8.0d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 2.0:
		tmp = 0.5
	else:
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(5.0 - Float64(8.0 / t)) / Float64(6.0 - Float64(8.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 2.0)
		tmp = 0.5;
	else
		tmp = (5.0 - (8.0 / t)) / (6.0 - (8.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 2.0], 0.5, N[(N[(5.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision] / N[(6.0 - N[(8.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.340000000000000024

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   6. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in t around inf 100.0%

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

   if -0.340000000000000024 < t < 2

   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%

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

      1. neg-mul-1 99.5%

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

      2. sub-neg 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in t around 0 99.5%

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

   if 2 < 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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in t around inf 97.6%

      \[\leadsto \frac{5 - \frac{8}{t}}{\color{blue}{6 - 8 \cdot \frac{1}{t}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.6%

         \[\leadsto \frac{5 - \frac{8}{t}}{6 - \color{blue}{\frac{8 \cdot 1}{t}}} \]

      2. metadata-eval 97.6%

         \[\leadsto \frac{5 - \frac{8}{t}}{6 - \frac{\color{blue}{8}}{t}} \]

   10. Simplified 97.6%

       \[\leadsto \frac{5 - \frac{8}{t}}{\color{blue}{6 - \frac{8}{t}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{8}{t}}{6 - \frac{8}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   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 98.3%

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

   6. Taylor expanded in t around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. distribute-neg-frac 98.3%

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

      5. metadata-eval 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in t around inf 98.4%

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

   if -0.340000000000000024 < 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 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%

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

      1. neg-mul-1 99.5%

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

      2. sub-neg 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in t around 0 99.5%

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

4. Final simplification 99.0%

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

Alternative 8: 59.5% 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 55.8%

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

6. Taylor expanded in t around 0 55.5%

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

   1. neg-mul-1 55.5%

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

   2. sub-neg 55.5%

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

8. Simplified 55.5%

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

9. Taylor expanded in t around 0 63.0%

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

10. Final simplification 63.0%

    \[\leadsto 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))))))