Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 11

Speedup: 1.0×

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, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{1}{2 + {\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (-
  1.0
  (/
   1.0
   (+
    2.0
    (pow
     (pow (+ 2.0 (/ (/ -2.0 t) (+ 1.0 (/ 1.0 t)))) 6.0)
     0.3333333333333333)))))

C

double code(double t) {
	return 1.0 - (1.0 / (2.0 + pow(pow((2.0 + ((-2.0 / t) / (1.0 + (1.0 / t)))), 6.0), 0.3333333333333333)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 1.0d0 - (1.0d0 / (2.0d0 + (((2.0d0 + (((-2.0d0) / t) / (1.0d0 + (1.0d0 / t)))) ** 6.0d0) ** 0.3333333333333333d0)))
end function

Java

public static double code(double t) {
	return 1.0 - (1.0 / (2.0 + Math.pow(Math.pow((2.0 + ((-2.0 / t) / (1.0 + (1.0 / t)))), 6.0), 0.3333333333333333)));
}

Python

def code(t):
	return 1.0 - (1.0 / (2.0 + math.pow(math.pow((2.0 + ((-2.0 / t) / (1.0 + (1.0 / t)))), 6.0), 0.3333333333333333)))

Julia

function code(t)
	return Float64(1.0 - Float64(1.0 / Float64(2.0 + ((Float64(2.0 + Float64(Float64(-2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) ^ 6.0) ^ 0.3333333333333333))))
end

MATLAB

function tmp = code(t)
	tmp = 1.0 - (1.0 / (2.0 + (((2.0 + ((-2.0 / t) / (1.0 + (1.0 / t)))) ^ 6.0) ^ 0.3333333333333333)));
end

Wolfram

code[t_] := N[(1.0 - N[(1.0 / N[(2.0 + N[Power[N[Power[N[(2.0 + N[(N[(-2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{\sqrt[3]{\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}} \]

   2. pow1/3 100.0%

      \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left(\left(\left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right) \cdot \left(\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\right)}^{0.3333333333333333}}} \]

4. Applied egg-rr 100.0%

   \[\leadsto 1 - \frac{1}{2 + \color{blue}{{\left({\left(2 + \frac{\frac{-2}{t}}{1 + \frac{1}{t}}\right)}^{6}\right)}^{0.3333333333333333}}} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.55:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(t \cdot \left(2 + -2 \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -2.55)
   0.8333333333333334
   (if (<= t 0.8)
     (-
      1.0
      (/
       1.0
       (+
        2.0
        (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (* t (+ 2.0 (* -2.0 t)))))))
     (+
      0.8333333333333334
      (*
       -1.0
       (/
        (+
         0.2222222222222222
         (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -2.55) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.8) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (t * (2.0 + (-2.0 * t))))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.55d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.8d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 + ((2.0d0 - ((2.0d0 / t) / (1.0d0 + (1.0d0 / t)))) * (t * (2.0d0 + ((-2.0d0) * t))))))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -2.55) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.8) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (t * (2.0 + (-2.0 * t))))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -2.55:
		tmp = 0.8333333333333334
	elif t <= 0.8:
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (t * (2.0 + (-2.0 * t))))))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -2.55)
		tmp = 0.8333333333333334;
	elseif (t <= 0.8)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t)))) * Float64(t * Float64(2.0 + Float64(-2.0 * t)))))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -2.55)
		tmp = 0.8333333333333334;
	elseif (t <= 0.8)
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (t * (2.0 + (-2.0 * t))))));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -2.55], 0.8333333333333334, If[LessEqual[t, 0.8], N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.5499999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -2.5499999999999998 < t < 0.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.6%

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

   if 0.80000000000000004 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

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

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(2 + -2 \cdot t\right)\\ \mathbf{if}\;t \leq -2.75:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.72:\\ \;\;\;\;1 - \frac{1}{2 + t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ 2.0 (* -2.0 t)))))
   (if (<= t -2.75)
     0.8333333333333334
     (if (<= t 0.72)
       (- 1.0 (/ 1.0 (+ 2.0 (* t_1 t_1))))
       (+
        0.8333333333333334
        (*
         -1.0
         (/
          (+
           0.2222222222222222
           (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
          t)))))))

C

double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -2.75) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.72) {
		tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (2.0d0 + ((-2.0d0) * t))
    if (t <= (-2.75d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.72d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 + (t_1 * t_1)))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (2.0 + (-2.0 * t));
	double tmp;
	if (t <= -2.75) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.72) {
		tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (2.0 + (-2.0 * t))
	tmp = 0
	if t <= -2.75:
		tmp = 0.8333333333333334
	elif t <= 0.72:
		tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(2.0 + Float64(-2.0 * t)))
	tmp = 0.0
	if (t <= -2.75)
		tmp = 0.8333333333333334;
	elseif (t <= 0.72)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(t_1 * t_1))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (2.0 + (-2.0 * t));
	tmp = 0.0;
	if (t <= -2.75)
		tmp = 0.8333333333333334;
	elseif (t <= 0.72)
		tmp = 1.0 - (1.0 / (2.0 + (t_1 * t_1)));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.75], 0.8333333333333334, If[LessEqual[t, 0.72], N[(1.0 - N[(1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.75

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -2.75 < t < 0.71999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.6%

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

   4. Taylor expanded in t around 0 99.6%

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

   if 0.71999999999999997 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

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

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 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]]

Derivation

1. Initial program 100.0%

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

Alternative 5: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.85:\\ \;\;\;\;1 - \frac{1}{2 + \left(t \cdot \left(2 + -2 \cdot t\right)\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -2.9)
   0.8333333333333334
   (if (<= t 0.85)
     (- 1.0 (/ 1.0 (+ 2.0 (* (* t (+ 2.0 (* -2.0 t))) (* 2.0 t)))))
     (+
      0.8333333333333334
      (*
       -1.0
       (/
        (+
         0.2222222222222222
         (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -2.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.85) {
		tmp = 1.0 - (1.0 / (2.0 + ((t * (2.0 + (-2.0 * t))) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.9d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.85d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 + ((t * (2.0d0 + ((-2.0d0) * t))) * (2.0d0 * t))))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -2.9) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.85) {
		tmp = 1.0 - (1.0 / (2.0 + ((t * (2.0 + (-2.0 * t))) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -2.9:
		tmp = 0.8333333333333334
	elif t <= 0.85:
		tmp = 1.0 - (1.0 / (2.0 + ((t * (2.0 + (-2.0 * t))) * (2.0 * t))))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -2.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.85)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(t * Float64(2.0 + Float64(-2.0 * t))) * Float64(2.0 * t)))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -2.9)
		tmp = 0.8333333333333334;
	elseif (t <= 0.85)
		tmp = 1.0 - (1.0 / (2.0 + ((t * (2.0 + (-2.0 * t))) * (2.0 * t))));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -2.9], 0.8333333333333334, If[LessEqual[t, 0.85], N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(t * N[(2.0 + N[(-2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.89999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -2.89999999999999991 < t < 0.849999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.4%

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

   if 0.849999999999999978 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

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

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;1 - \frac{1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \frac{0.04938271604938271}{t}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -3.15)
   0.8333333333333334
   (if (<= t 0.66)
     (- 1.0 (/ 1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (+
      0.8333333333333334
      (*
       -1.0
       (/
        (+
         0.2222222222222222
         (* -1.0 (/ (+ 0.037037037037037035 (/ 0.04938271604938271 t)) t)))
        t))))))

C

double code(double t) {
	double tmp;
	if (t <= -3.15) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.15d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.66d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    else
        tmp = 0.8333333333333334d0 + ((-1.0d0) * ((0.2222222222222222d0 + ((-1.0d0) * ((0.037037037037037035d0 + (0.04938271604938271d0 / t)) / t))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -3.15) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -3.15:
		tmp = 0.8333333333333334
	elif t <= 0.66:
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -3.15)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	else
		tmp = Float64(0.8333333333333334 + Float64(-1.0 * Float64(Float64(0.2222222222222222 + Float64(-1.0 * Float64(Float64(0.037037037037037035 + Float64(0.04938271604938271 / t)) / t))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -3.15)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 0.8333333333333334 + (-1.0 * ((0.2222222222222222 + (-1.0 * ((0.037037037037037035 + (0.04938271604938271 / t)) / t))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -3.15], 0.8333333333333334, If[LessEqual[t, 0.66], N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 + N[(-1.0 * N[(N[(0.2222222222222222 + N[(-1.0 * N[(N[(0.037037037037037035 + N[(0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.14999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -3.14999999999999991 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.3%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.8%

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

   4. Taylor expanded in t around 0 98.8%

      \[\leadsto 0.8333333333333334 + -1 \cdot \frac{0.2222222222222222 + -1 \cdot \frac{0.037037037037037035 + \color{blue}{\frac{0.04938271604938271}{t}}}{t}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.44:\\ \;\;\;\;1 - \frac{1}{2 + \left(2 \cdot t\right) \cdot \left(2 \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -3.15)
   0.8333333333333334
   (if (<= t 0.44)
     (- 1.0 (/ 1.0 (+ 2.0 (* (* 2.0 t) (* 2.0 t)))))
     (-
      0.8333333333333334
      (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -3.15) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.44) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} 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 <= (-3.15d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.44d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 + ((2.0d0 * t) * (2.0d0 * t))))
    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 <= -3.15) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.44) {
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -3.15:
		tmp = 0.8333333333333334
	elif t <= 0.44:
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))))
	else:
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -3.15)
		tmp = 0.8333333333333334;
	elseif (t <= 0.44)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 + Float64(Float64(2.0 * t) * Float64(2.0 * t)))));
	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 <= -3.15)
		tmp = 0.8333333333333334;
	elseif (t <= 0.44)
		tmp = 1.0 - (1.0 / (2.0 + ((2.0 * t) * (2.0 * t))));
	else
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -3.15], 0.8333333333333334, If[LessEqual[t, 0.44], N[(1.0 - N[(1.0 / N[(2.0 + N[(N[(2.0 * t), $MachinePrecision] * N[(2.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.14999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -3.14999999999999991 < t < 0.440000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.4%

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

   4. Taylor expanded in t around 0 99.3%

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

   if 0.440000000000000002 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. un-div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unsub-neg 98.1%

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

   7. Applied egg-rr 98.1%

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

Alternative 8: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.23:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.23)
     0.5
     (-
      0.8333333333333334
      (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} 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 <= (-0.33d0)) then
        tmp = 0.8333333333333334d0
    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 <= -0.33) {
		tmp = 0.8333333333333334;
	} 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 <= -0.33:
		tmp = 0.8333333333333334
	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 <= -0.33)
		tmp = 0.8333333333333334;
	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 <= -0.33)
		tmp = 0.8333333333333334;
	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, -0.33], 0.8333333333333334, 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 < -0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.23000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

   if 0.23000000000000001 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. un-div-inv 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unsub-neg 98.1%

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

   7. Applied egg-rr 98.1%

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

Alternative 9: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.33:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.33)
   0.8333333333333334
   (if (<= t 0.66) 0.5 (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.33) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	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 <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    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 <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.33:
		tmp = 0.8333333333333334
	elif t <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.33)
		tmp = 0.8333333333333334;
	elseif (t <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.33], 0.8333333333333334, If[LessEqual[t, 0.66], 0.5, N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.330000000000000016

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

   if -0.330000000000000016 < t < 0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

   if 0.660000000000000031 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 97.5%

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

   4. Taylor expanded in t around 0 97.5%

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

Alternative 10: 98.7% 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%

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

   3. Taylor expanded in t around inf 98.0%

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

   if -0.330000000000000016 < t < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0 99.1%

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

Alternative 11: 59.1% 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%

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

3. Taylor expanded in t around 0 59.4%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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)))))))))