Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 14.1s

Alternatives: 7

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 99.6%

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

   1. associate-/l* 99.6%

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

   2. associate-/l* 99.6%

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

   3. swap-sqr 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate-/l* 99.6%

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

   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: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.49:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{\frac{-12}{t} + 8}{t}}{\frac{-8 + \frac{12}{t}}{t} + 6}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.35)
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (if (<= t 0.49)
     (/ (+ (* 4.0 (* t t)) 1.0) 2.0)
     (/ (- 5.0 (/ (+ (/ -12.0 t) 8.0) t)) (+ (/ (+ -8.0 (/ 12.0 t)) t) 6.0)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.35) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else if (t <= 0.49) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = (5.0 - (((-12.0 / t) + 8.0) / t)) / (((-8.0 + (12.0 / t)) / t) + 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.35d0)) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else if (t <= 0.49d0) then
        tmp = ((4.0d0 * (t * t)) + 1.0d0) / 2.0d0
    else
        tmp = (5.0d0 - ((((-12.0d0) / t) + 8.0d0) / t)) / ((((-8.0d0) + (12.0d0 / t)) / t) + 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.35) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else if (t <= 0.49) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = (5.0 - (((-12.0 / t) + 8.0) / t)) / (((-8.0 + (12.0 / t)) / t) + 6.0);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.35:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	elif t <= 0.49:
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0
	else:
		tmp = (5.0 - (((-12.0 / t) + 8.0) / t)) / (((-8.0 + (12.0 / t)) / t) + 6.0)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.35)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	elseif (t <= 0.49)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * t)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(5.0 - Float64(Float64(Float64(-12.0 / t) + 8.0) / t)) / Float64(Float64(Float64(-8.0 + Float64(12.0 / t)) / t) + 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.35)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	elseif (t <= 0.49)
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	else
		tmp = (5.0 - (((-12.0 / t) + 8.0) / t)) / (((-8.0 + (12.0 / t)) / t) + 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.35], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.49], N[(N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(5.0 - N[(N[(N[(-12.0 / t), $MachinePrecision] + 8.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-8.0 + N[(12.0 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.34999999999999998

   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 97.1%

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

      1. mul-1-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      2. unsub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]

      3. mul-1-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}} \]

      4. unsub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}} \]

      5. sub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}} \]

      6. associate-*r/ 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}} \]

      7. metadata-eval 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}} \]

      8. distribute-neg-frac 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}} \]

      9. metadata-eval 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}} \]

   7. Simplified 97.1%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   9. Step-by-step derivation

      1. mul-1-neg 97.5%

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

      2. unsub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. unsub-neg 97.5%

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

      5. associate-*r/ 97.5%

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

      6. metadata-eval 97.5%

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

   10. Simplified 97.5%

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

   if -0.34999999999999998 < t < 0.48999999999999999

   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 99.6%

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

   6. Taylor expanded in t around 0 99.6%

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

   7. Taylor expanded in t around 0 99.6%

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

   if 0.48999999999999999 < t

   1. Initial program 98.6%

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

      1. associate-/l* 98.6%

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

      2. associate-/l* 98.6%

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

      3. swap-sqr 98.6%

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

      4. metadata-eval 98.6%

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

      5. associate-/l* 98.6%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) + \left(-8 \cdot \frac{1}{t}\right)}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(-8 \cdot \frac{1}{t}\right) + \left(6 + \frac{12}{{t}^{2}}\right)}} \]

      3. associate-+r+ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(\left(-8 \cdot \frac{1}{t}\right) + 6\right) + \frac{12}{{t}^{2}}}} \]

      4. +-commutative 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)} + \frac{12}{{t}^{2}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)} + \frac{12}{{t}^{2}}} \]

      6. associate--r- 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)} \]

      9. unpow2 100.0%

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

      10. associate-/r* 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-*r/ 100.0%

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

      13. div-sub 100.0%

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

      14. unsub-neg 100.0%

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

      15. mul-1-neg 100.0%

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

      16. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. associate--r- 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

      9. unpow2 100.0%

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

      10. associate-/r* 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-*r/ 100.0%

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

      13. div-sub 100.0%

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

      14. sub-neg 100.0%

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

      15. associate-*r/ 100.0%

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

      16. metadata-eval 100.0%

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

      17. distribute-neg-frac 100.0%

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

      18. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.49:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 - \frac{\frac{-12}{t} + 8}{t}}{\frac{-8 + \frac{12}{t}}{t} + 6}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.195 \lor \neg \left(t \leq 0.17\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot t + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.195) (not (<= t 0.17)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (/ (+ (* 4.0 t) 1.0) 2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.195) || !(t <= 0.17)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = ((4.0 * t) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.195d0)) .or. (.not. (t <= 0.17d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = ((4.0d0 * t) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.195) || !(t <= 0.17)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = ((4.0 * t) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.195) or not (t <= 0.17):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = ((4.0 * t) + 1.0) / 2.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.195) || !(t <= 0.17))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(Float64(Float64(4.0 * t) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.195) || ~((t <= 0.17)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = ((4.0 * t) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.195], N[Not[LessEqual[t, 0.17]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * t), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.19500000000000001 or 0.170000000000000012 < t

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. associate-/l* 99.3%

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

      3. swap-sqr 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-/l* 99.3%

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

      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%

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

      1. sub-neg 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) + \left(-8 \cdot \frac{1}{t}\right)}} \]

      2. +-commutative 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(-8 \cdot \frac{1}{t}\right) + \left(6 + \frac{12}{{t}^{2}}\right)}} \]

      3. associate-+r+ 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(\left(-8 \cdot \frac{1}{t}\right) + 6\right) + \frac{12}{{t}^{2}}}} \]

      4. +-commutative 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)} + \frac{12}{{t}^{2}}} \]

      5. sub-neg 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)} + \frac{12}{{t}^{2}}} \]

      6. associate--r- 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)}} \]

      7. associate-*r/ 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)} \]

      8. metadata-eval 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)} \]

      9. unpow2 98.3%

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

      10. associate-/r* 98.3%

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

      11. metadata-eval 98.3%

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

      12. associate-*r/ 98.3%

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

      13. div-sub 98.3%

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

      14. unsub-neg 98.3%

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

      15. mul-1-neg 98.3%

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

      16. +-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. distribute-neg-frac 98.5%

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

      7. metadata-eval 98.5%

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

   10. Simplified 98.5%

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

   if -0.19500000000000001 < t < 0.170000000000000012

   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 99.6%

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

   6. Taylor expanded in t around 0 99.6%

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

   7. Taylor expanded in t around inf 98.2%

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

      1. *-commutative 98.2%

         \[\leadsto \frac{1 + \color{blue}{t \cdot 4}}{2} \]

   9. Simplified 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.195 \lor \neg \left(t \leq 0.17\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot t + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.215\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.62) (not (<= t 0.215)))
   (-
    0.8333333333333334
    (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t))
   (/ (+ (* 4.0 (* t t)) 1.0) 2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.62) || !(t <= 0.215)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.62d0)) .or. (.not. (t <= 0.215d0))) then
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    else
        tmp = ((4.0d0 * (t * t)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.62) || !(t <= 0.215)) {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	} else {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.62) or not (t <= 0.215):
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	else:
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.62) || !(t <= 0.215))
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(t * t)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.62) || ~((t <= 0.215)))
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	else
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.62], N[Not[LessEqual[t, 0.215]], $MachinePrecision]], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.619999999999999996 or 0.214999999999999997 < t

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. associate-/l* 99.3%

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

      3. swap-sqr 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-/l* 99.3%

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

      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%

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

      1. sub-neg 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) + \left(-8 \cdot \frac{1}{t}\right)}} \]

      2. +-commutative 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(-8 \cdot \frac{1}{t}\right) + \left(6 + \frac{12}{{t}^{2}}\right)}} \]

      3. associate-+r+ 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(\left(-8 \cdot \frac{1}{t}\right) + 6\right) + \frac{12}{{t}^{2}}}} \]

      4. +-commutative 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)} + \frac{12}{{t}^{2}}} \]

      5. sub-neg 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)} + \frac{12}{{t}^{2}}} \]

      6. associate--r- 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)}} \]

      7. associate-*r/ 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)} \]

      8. metadata-eval 98.3%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)} \]

      9. unpow2 98.3%

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

      10. associate-/r* 98.3%

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

      11. metadata-eval 98.3%

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

      12. associate-*r/ 98.3%

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

      13. div-sub 98.3%

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

      14. unsub-neg 98.3%

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

      15. mul-1-neg 98.3%

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

      16. +-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in t around -inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. associate-*r/ 98.5%

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

      5. metadata-eval 98.5%

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

      6. distribute-neg-frac 98.5%

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

      7. metadata-eval 98.5%

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

   10. Simplified 98.5%

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

   if -0.619999999999999996 < t < 0.214999999999999997

   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 99.6%

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

   6. Taylor expanded in t around 0 99.6%

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

   7. Taylor expanded in t around 0 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.62 \lor \neg \left(t \leq 0.215\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.215:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.35)
   (+
    0.8333333333333334
    (/
     (-
      (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
      0.2222222222222222)
     t))
   (if (<= t 0.215)
     (/ (+ (* 4.0 (* t t)) 1.0) 2.0)
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.35) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else if (t <= 0.215) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.35d0)) then
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    else if (t <= 0.215d0) then
        tmp = ((4.0d0 * (t * t)) + 1.0d0) / 2.0d0
    else
        tmp = 0.8333333333333334d0 - ((0.2222222222222222d0 + ((-0.037037037037037035d0) / t)) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.35) {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	} else if (t <= 0.215) {
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.35:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	elif t <= 0.215:
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.35)
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	elseif (t <= 0.215)
		tmp = Float64(Float64(Float64(4.0 * Float64(t * t)) + 1.0) / 2.0);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 + Float64(-0.037037037037037035 / t)) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.35)
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	elseif (t <= 0.215)
		tmp = ((4.0 * (t * t)) + 1.0) / 2.0;
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.35], N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.215], N[(N[(N[(4.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 + N[(-0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.34999999999999998

   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 97.1%

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

      1. mul-1-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 + \color{blue}{\left(-\frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}\right)}} \]

      2. unsub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 + -1 \cdot \frac{12 - 16 \cdot \frac{1}{t}}{t}}{t}}} \]

      3. mul-1-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 + \color{blue}{\left(-\frac{12 - 16 \cdot \frac{1}{t}}{t}\right)}}{t}} \]

      4. unsub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{\color{blue}{8 - \frac{12 - 16 \cdot \frac{1}{t}}{t}}}{t}} \]

      5. sub-neg 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{\color{blue}{12 + \left(-16 \cdot \frac{1}{t}\right)}}{t}}{t}} \]

      6. associate-*r/ 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \left(-\color{blue}{\frac{16 \cdot 1}{t}}\right)}{t}}{t}} \]

      7. metadata-eval 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \left(-\frac{\color{blue}{16}}{t}\right)}{t}}{t}} \]

      8. distribute-neg-frac 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \color{blue}{\frac{-16}{t}}}{t}}{t}} \]

      9. metadata-eval 97.1%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \frac{8 - \frac{12 + \frac{\color{blue}{-16}}{t}}{t}}{t}} \]

   7. Simplified 97.1%

      \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \frac{8 - \frac{12 + \frac{-16}{t}}{t}}{t}}} \]

   8. Taylor expanded in t around -inf 97.5%

      \[\leadsto \color{blue}{0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + 0.04938271604938271 \cdot \frac{1}{t}}{t}}{t}} \]
   9. Step-by-step derivation

      1. mul-1-neg 97.5%

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

      2. unsub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. unsub-neg 97.5%

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

      5. associate-*r/ 97.5%

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

      6. metadata-eval 97.5%

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

   10. Simplified 97.5%

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

   if -0.34999999999999998 < t < 0.214999999999999997

   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 99.6%

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

   6. Taylor expanded in t around 0 99.6%

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

   7. Taylor expanded in t around 0 99.6%

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

   if 0.214999999999999997 < t

   1. Initial program 98.6%

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

      1. associate-/l* 98.6%

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

      2. associate-/l* 98.6%

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

      3. swap-sqr 98.6%

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

      4. metadata-eval 98.6%

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

      5. associate-/l* 98.6%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \frac{12}{{t}^{2}}\right) + \left(-8 \cdot \frac{1}{t}\right)}} \]

      2. +-commutative 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(-8 \cdot \frac{1}{t}\right) + \left(6 + \frac{12}{{t}^{2}}\right)}} \]

      3. associate-+r+ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(\left(-8 \cdot \frac{1}{t}\right) + 6\right) + \frac{12}{{t}^{2}}}} \]

      4. +-commutative 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 + \left(-8 \cdot \frac{1}{t}\right)\right)} + \frac{12}{{t}^{2}}} \]

      5. sub-neg 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{\left(6 - 8 \cdot \frac{1}{t}\right)} + \frac{12}{{t}^{2}}} \]

      6. associate--r- 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{\color{blue}{6 - \left(8 \cdot \frac{1}{t} - \frac{12}{{t}^{2}}\right)}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\color{blue}{\frac{8 \cdot 1}{t}} - \frac{12}{{t}^{2}}\right)} \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{1 + 4 \cdot \left(\frac{t}{1 + t} \cdot \frac{t}{1 + t}\right)}{6 - \left(\frac{\color{blue}{8}}{t} - \frac{12}{{t}^{2}}\right)} \]

      9. unpow2 100.0%

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

      10. associate-/r* 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-*r/ 100.0%

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

      13. div-sub 100.0%

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

      14. unsub-neg 100.0%

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

      15. mul-1-neg 100.0%

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

      16. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in t around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.35:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.215:\\ \;\;\;\;\frac{4 \cdot \left(t \cdot t\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.18 \lor \neg \left(t \leq 0.38\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot t + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.18) (not (<= t 0.38)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (/ (+ (* 4.0 t) 1.0) 2.0)))

C

double code(double t) {
	double tmp;
	if ((t <= -0.18) || !(t <= 0.38)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = ((4.0 * t) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.18d0)) .or. (.not. (t <= 0.38d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = ((4.0d0 * t) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.18) || !(t <= 0.38)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = ((4.0 * t) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.18) or not (t <= 0.38):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = ((4.0 * t) + 1.0) / 2.0
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.18) || !(t <= 0.38))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(Float64(4.0 * t) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.18) || ~((t <= 0.38)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = ((4.0 * t) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.18], N[Not[LessEqual[t, 0.38]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * t), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.17999999999999999 or 0.38 < t

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

      2. associate-/l* 99.3%

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

      3. swap-sqr 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-/l* 99.3%

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

      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 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in t around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. metadata-eval 98.0%

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

   10. Simplified 98.0%

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

   if -0.17999999999999999 < 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 0 99.6%

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

   6. Taylor expanded in t around 0 99.6%

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

   7. Taylor expanded in t around inf 98.2%

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

      1. *-commutative 98.2%

         \[\leadsto \frac{1 + \color{blue}{t \cdot 4}}{2} \]

   9. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.18 \lor \neg \left(t \leq 0.38\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot t + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 0.8333333333333334)

C

double code(double t) {
	return 0.8333333333333334;
}

Fortran

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

Java

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

Python

def code(t):
	return 0.8333333333333334

Julia

function code(t)
	return 0.8333333333333334
end

MATLAB

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

Wolfram

code[t_] := 0.8333333333333334

Derivation

1. Initial program 99.6%

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

   1. associate-/l* 99.6%

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

   2. associate-/l* 99.6%

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

   3. swap-sqr 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate-/l* 99.6%

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

   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 61.1%

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

6. Taylor expanded in t around inf 62.2%

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

7. Final simplification 62.2%

   \[\leadsto 0.8333333333333334 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))))))