Kahan p13 Example 1

Percentage Accurate: 99.9% → 100.0%

Time: 18.0s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 100.0%

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

   2. associate-/l* 100.0%

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

   3. swap-sqr 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l* 100.0%

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

   6. associate-/l* 100.0%

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

   7. swap-sqr 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-12}{t} + 8}{t}\\ \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;\frac{5 - t\_1}{6 - t\_1}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (+ (/ -12.0 t) 8.0) t)))
   (if (<= t -0.42)
     (/ (- 5.0 t_1) (- 6.0 t_1))
     (if (<= t 0.68)
       0.5
       (+
        0.8333333333333334
        (/
         (-
          (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
          0.2222222222222222)
         t))))))

C

double code(double t) {
	double t_1 = ((-12.0 / t) + 8.0) / t;
	double tmp;
	if (t <= -0.42) {
		tmp = (5.0 - t_1) / (6.0 - t_1);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((-12.0d0) / t) + 8.0d0) / t
    if (t <= (-0.42d0)) then
        tmp = (5.0d0 - t_1) / (6.0d0 - t_1)
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = ((-12.0 / t) + 8.0) / t;
	double tmp;
	if (t <= -0.42) {
		tmp = (5.0 - t_1) / (6.0 - t_1);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	t_1 = ((-12.0 / t) + 8.0) / t
	tmp = 0
	if t <= -0.42:
		tmp = (5.0 - t_1) / (6.0 - t_1)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(Float64(-12.0 / t) + 8.0) / t)
	tmp = 0.0
	if (t <= -0.42)
		tmp = Float64(Float64(5.0 - t_1) / Float64(6.0 - t_1));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = ((-12.0 / t) + 8.0) / t;
	tmp = 0.0;
	if (t <= -0.42)
		tmp = (5.0 - t_1) / (6.0 - t_1);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(N[(-12.0 / t), $MachinePrecision] + 8.0), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -0.42], N[(N[(5.0 - t$95$1), $MachinePrecision] / N[(6.0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.419999999999999984

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. associate-/r* 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. mul-1-neg 99.9%

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

      11. sub-neg 99.9%

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

      12. metadata-eval 99.9%

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

      13. distribute-lft-in 99.9%

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

      14. neg-mul-1 99.9%

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

      15. metadata-eval 99.9%

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

      16. +-commutative 99.9%

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

      17. sub-neg 99.9%

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

      18. distribute-neg-frac 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. unpow2 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*r/ 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. div-sub 100.0%

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

      11. remove-double-neg 100.0%

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

      12. mul-1-neg 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

      15. distribute-lft-in 100.0%

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

      16. neg-mul-1 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-commutative 100.0%

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

      19. sub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -0.419999999999999984 < t < 0.680000000000000049

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

      3. mul-1-neg 99.3%

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

      4. unsub-neg 99.3%

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

      5. associate-*r/ 99.3%

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

      6. metadata-eval 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.42:\\ \;\;\;\;\frac{5 - \frac{\frac{-12}{t} + 8}{t}}{6 - \frac{\frac{-12}{t} + 8}{t}}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.48)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.68)
     0.5
     (+
      0.8333333333333334
      (/
       (-
        (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)
        0.2222222222222222)
       t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.68d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 + ((((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t) - 0.2222222222222222d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.68) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.68:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 + Float64(Float64(Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t) - 0.2222222222222222) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.48)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.68)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + ((((0.037037037037037035 + (0.04938271604938271 / t)) / t) - 0.2222222222222222) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.68], 0.5, N[(0.8333333333333334 + N[(N[(N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - 0.2222222222222222), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.47999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.47999999999999998 < t < 0.680000000000000049

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 0.680000000000000049 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

      3. mul-1-neg 99.3%

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

      4. unsub-neg 99.3%

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

      5. associate-*r/ 99.3%

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

      6. metadata-eval 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.68:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{\frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t} - 0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.48)
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   (if (<= t 0.23)
     0.5
     (-
      0.8333333333333334
      (/ (+ 0.2222222222222222 (/ -0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} 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 <= (-0.48d0)) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else if (t <= 0.23d0) then
        tmp = 0.5d0
    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 <= -0.48) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else if (t <= 0.23) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.48:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	elif t <= 0.23:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.48)
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	elseif (t <= 0.23)
		tmp = 0.5;
	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 <= -0.48)
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	elseif (t <= 0.23)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 + (-0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.48], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.23], 0.5, 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 < -0.47999999999999998

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -0.47999999999999998 < t < 0.23000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

   if 0.23000000000000001 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around -inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. associate-*r/ 99.2%

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

      5. metadata-eval 99.2%

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

      6. distribute-neg-frac 99.2%

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

      7. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 + \frac{-0.037037037037037035}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.48) (not (<= t 0.68)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.48d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.48) || !(t <= 0.68)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.48) or not (t <= 0.68):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.48) || !(t <= 0.68))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.48) || ~((t <= 0.68)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.48], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.47999999999999998 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -0.47999999999999998 < t < 0.680000000000000049

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.48 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.330000000000000016 or 1 < t

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 97.6%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/l* 100.0%

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

      3. swap-sqr 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-/l* 100.0%

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

      6. associate-/l* 100.0%

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

      7. swap-sqr 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 98.8%

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

4. Final simplification 98.2%

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

Alternative 7: 58.9% 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 57.7%

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

6. Final simplification 57.7%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))))