Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{1 + t}\\ 1 + \frac{-1}{6 + t\_1 \cdot \left(t\_1 + -4\right)} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (+ 1.0 t))))
   (+ 1.0 (/ -1.0 (+ 6.0 (* t_1 (+ t_1 -4.0)))))))

C

double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 2.0d0 / (1.0d0 + t)
    code = 1.0d0 + ((-1.0d0) / (6.0d0 + (t_1 * (t_1 + (-4.0d0)))))
end function

Java

public static double code(double t) {
	double t_1 = 2.0 / (1.0 + t);
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
}

Python

def code(t):
	t_1 = 2.0 / (1.0 + t)
	return 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))))

Julia

function code(t)
	t_1 = Float64(2.0 / Float64(1.0 + t))
	return Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(t_1 * Float64(t_1 + -4.0)))))
end

MATLAB

function tmp = code(t)
	t_1 = 2.0 / (1.0 + t);
	tmp = 1.0 + (-1.0 / (6.0 + (t_1 * (t_1 + -4.0))));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(2.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(-1.0 / N[(6.0 + N[(t$95$1 * N[(t$95$1 + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.59 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 + \left(1 + t\right) \cdot \left(t \cdot 4 - 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.59) (not (<= t 0.68)))
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (+ 1.0 (/ -1.0 (+ 6.0 (* (+ 1.0 t) (- (* t 4.0) 4.0)))))))

C

double code(double t) {
	double tmp;
	if ((t <= -0.59) || !(t <= 0.68)) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.59d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((1.0d0 + t) * ((t * 4.0d0) - 4.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.59) || !(t <= 0.68)) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.59) or not (t <= 0.68):
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	else:
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.59) || !(t <= 0.68))
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(1.0 + t) * Float64(Float64(t * 4.0) - 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.59) || ~((t <= 0.68)))
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	else
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.59], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(1.0 + t), $MachinePrecision] * N[(N[(t * 4.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.589999999999999969 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.4%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 99.4%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 99.4%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 99.4%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.589999999999999969 < t < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. +-commutative 99.0%

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

      5. *-commutative 99.0%

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

      6. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft1-in 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in t around 0 99.1%

       \[\leadsto 1 + \frac{-1}{6 + \left(t + 1\right) \cdot \color{blue}{\left(4 \cdot t - 4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.59 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 + \left(1 + t\right) \cdot \left(t \cdot 4 - 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.59:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{6 + \left(1 + t\right) \cdot \left(t \cdot 4 - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.59)
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   (if (<= t 0.76)
     (+ 1.0 (/ -1.0 (+ 6.0 (* (+ 1.0 t) (- (* t 4.0) 4.0)))))
     (+ 1.0 (/ -1.0 (+ 6.0 (/ (- 8.0 (/ 4.0 t)) (- -1.0 t))))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.59) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.76) {
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.59d0)) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else if (t <= 0.76d0) then
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((1.0d0 + t) * ((t * 4.0d0) - 4.0d0))))
    else
        tmp = 1.0d0 + ((-1.0d0) / (6.0d0 + ((8.0d0 - (4.0d0 / t)) / ((-1.0d0) - t))))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.59) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else if (t <= 0.76) {
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	} else {
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.59:
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	elif t <= 0.76:
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))))
	else:
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.59)
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	elseif (t <= 0.76)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(1.0 + t) * Float64(Float64(t * 4.0) - 4.0)))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(8.0 - Float64(4.0 / t)) / Float64(-1.0 - t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.59)
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	elseif (t <= 0.76)
		tmp = 1.0 + (-1.0 / (6.0 + ((1.0 + t) * ((t * 4.0) - 4.0))));
	else
		tmp = 1.0 + (-1.0 / (6.0 + ((8.0 - (4.0 / t)) / (-1.0 - t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.59], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.76], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(1.0 + t), $MachinePrecision] * N[(N[(t * 4.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(8.0 - N[(4.0 / t), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.589999999999999969

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 100.0%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 100.0%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 100.0%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 100.0%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.589999999999999969 < t < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. +-commutative 99.0%

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

      5. *-commutative 99.0%

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

      6. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft1-in 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in t around 0 99.1%

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

   if 0.76000000000000001 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*l/ 100.0%

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

      2. frac-2neg 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. distribute-neg-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. unsub-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. unsub-neg 100.0%

          \[\leadsto 1 + \frac{-1}{6 + \frac{8 + \frac{-4}{t + 1}}{\color{blue}{-1 - t}}} \]

   8. Simplified 100.0%

      \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{8 + \frac{-4}{t + 1}}{-1 - t}}} \]

   9. Taylor expanded in t around inf 98.5%

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

       1. associate-*r/ 98.5%

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

       2. metadata-eval 98.5%

          \[\leadsto 1 + \frac{-1}{6 + \frac{8 - \frac{\color{blue}{4}}{t}}{-1 - t}} \]

   11. Simplified 98.5%

       \[\leadsto 1 + \frac{-1}{6 + \frac{\color{blue}{8 - \frac{4}{t}}}{-1 - t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.59:\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{elif}\;t \leq 0.76:\\ \;\;\;\;1 + \frac{-1}{6 + \left(1 + t\right) \cdot \left(t \cdot 4 - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{6 + \frac{8 - \frac{4}{t}}{-1 - t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.68)))
   (+ 1.0 (+ -0.16666666666666666 (/ -0.2222222222222222 t)))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.68d0))) then
        tmp = 1.0d0 + ((-0.16666666666666666d0) + ((-0.2222222222222222d0) / t))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.68)) {
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.68):
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t))
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.68))
		tmp = Float64(1.0 + Float64(-0.16666666666666666 + Float64(-0.2222222222222222 / t)));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.68)))
		tmp = 1.0 + (-0.16666666666666666 + (-0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.68]], $MachinePrecision]], N[(1.0 + N[(-0.16666666666666666 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if t < -0.48999999999999999 or 0.680000000000000049 < t

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 99.4%

      \[\leadsto 1 + \color{blue}{\left(-\left(0.16666666666666666 + 0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-neg-in 99.4%

         \[\leadsto 1 + \color{blue}{\left(\left(-0.16666666666666666\right) + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right)} \]

      2. metadata-eval 99.4%

         \[\leadsto 1 + \left(\color{blue}{-0.16666666666666666} + \left(-0.2222222222222222 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\color{blue}{\frac{0.2222222222222222 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \left(-\frac{\color{blue}{0.2222222222222222}}{t}\right)\right) \]

      5. distribute-neg-frac 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \color{blue}{\frac{-0.2222222222222222}{t}}\right) \]

      6. metadata-eval 99.4%

         \[\leadsto 1 + \left(-0.16666666666666666 + \frac{\color{blue}{-0.2222222222222222}}{t}\right) \]

   7. Simplified 99.4%

      \[\leadsto 1 + \color{blue}{\left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)} \]

   if -0.48999999999999999 < t < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. +-commutative 99.0%

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

      5. *-commutative 99.0%

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

      6. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft1-in 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in t around 0 99.1%

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

   11. Taylor expanded in t around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.68\right):\\ \;\;\;\;1 + \left(-0.16666666666666666 + \frac{-0.2222222222222222}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{6 + \frac{8 + \frac{-4}{1 + t}}{-1 - t}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (+ 1.0 (/ -1.0 (+ 6.0 (/ (+ 8.0 (/ -4.0 (+ 1.0 t))) (- -1.0 t))))))

C

double code(double t) {
	return 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / (1.0 + t))) / (-1.0 - t))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 + ((-1.0d0) / (6.0d0 + ((8.0d0 + ((-4.0d0) / (1.0d0 + t))) / ((-1.0d0) - t))))
end function

Java

public static double code(double t) {
	return 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / (1.0 + t))) / (-1.0 - t))));
}

Python

def code(t):
	return 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / (1.0 + t))) / (-1.0 - t))))

Julia

function code(t)
	return Float64(1.0 + Float64(-1.0 / Float64(6.0 + Float64(Float64(8.0 + Float64(-4.0 / Float64(1.0 + t))) / Float64(-1.0 - t)))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 + (-1.0 / (6.0 + ((8.0 + (-4.0 / (1.0 + t))) / (-1.0 - t))));
end

Wolfram

code[t_] := N[(1.0 + N[(-1.0 / N[(6.0 + N[(N[(8.0 + N[(-4.0 / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. associate-*l/ 100.0%

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

   2. frac-2neg 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. metadata-eval 100.0%

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

   6. distribute-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. distribute-neg-in 100.0%

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

   2. metadata-eval 100.0%

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

   3. unsub-neg 100.0%

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

   4. unsub-neg 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. distribute-neg-frac 100.0%

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

   8. metadata-eval 100.0%

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

   9. +-commutative 100.0%

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

   10. unsub-neg 100.0%

       \[\leadsto 1 + \frac{-1}{6 + \frac{8 + \frac{-4}{t + 1}}{\color{blue}{-1 - t}}} \]

8. Simplified 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \color{blue}{\frac{8 + \frac{-4}{t + 1}}{-1 - t}}} \]

9. Final simplification 100.0%

   \[\leadsto 1 + \frac{-1}{6 + \frac{8 + \frac{-4}{1 + t}}{-1 - t}} \]
10. Add Preprocessing

Alternative 6: 98.5% accurate, 2.6× 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%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 98.5%

      \[\leadsto 1 + \color{blue}{-0.16666666666666666} \]

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. +-commutative 99.0%

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

      5. *-commutative 99.0%

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

      6. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l/ 99.0%

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

      3. metadata-eval 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*r/ 99.0%

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

      6. metadata-eval 99.0%

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

      7. distribute-lft1-in 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in t around 0 99.1%

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

   11. Taylor expanded in t around 0 99.0%

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

4. Final simplification 98.8%

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

Alternative 7: 59.8% accurate, 29.0× 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%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around 0 61.8%

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

   1. sub-neg 61.8%

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

   2. metadata-eval 61.8%

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

   3. distribute-lft-in 61.8%

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

   4. +-commutative 61.8%

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

   5. *-commutative 61.8%

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

   6. +-commutative 61.8%

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

7. Applied egg-rr 61.8%

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

   1. *-commutative 61.8%

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

   2. associate-*l/ 61.8%

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

   3. metadata-eval 61.8%

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

   4. associate-*l* 61.8%

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

   5. associate-*r/ 61.8%

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

   6. metadata-eval 61.8%

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

   7. distribute-lft1-in 61.8%

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

9. Simplified 61.8%

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

10. Taylor expanded in t around 0 62.6%

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

11. Taylor expanded in t around 0 61.8%

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

12. Final simplification 61.8%

    \[\leadsto 0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))