Kahan p13 Example 3

Percentage Accurate: 100.0% → 100.0%

Time: 13.3s

Alternatives: 9

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(1.0 + N[(1.0 / N[(N[(N[(2.0 + N[(N[(2.0 / t), $MachinePrecision] / N[(-1.0 - N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] - 2.0), $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. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 - \frac{1}{2 - \left(t \cdot \left(2 + t \cdot -2\right)\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -2.6)
   0.8333333333333334
   (if (<= t 0.8)
     (-
      1.0
      (/
       1.0
       (-
        2.0
        (* (* t (+ 2.0 (* t -2.0))) (- (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))) 2.0)))))
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -2.6) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.8) {
		tmp = 1.0 - (1.0 / (2.0 - ((t * (2.0 + (t * -2.0))) * (((2.0 / t) / (1.0 + (1.0 / t))) - 2.0))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.6d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.8d0) then
        tmp = 1.0d0 - (1.0d0 / (2.0d0 - ((t * (2.0d0 + (t * (-2.0d0)))) * (((2.0d0 / t) / (1.0d0 + (1.0d0 / t))) - 2.0d0))))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -2.6) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.8) {
		tmp = 1.0 - (1.0 / (2.0 - ((t * (2.0 + (t * -2.0))) * (((2.0 / t) / (1.0 + (1.0 / t))) - 2.0))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -2.6:
		tmp = 0.8333333333333334
	elif t <= 0.8:
		tmp = 1.0 - (1.0 / (2.0 - ((t * (2.0 + (t * -2.0))) * (((2.0 / t) / (1.0 + (1.0 / t))) - 2.0))))
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -2.6)
		tmp = 0.8333333333333334;
	elseif (t <= 0.8)
		tmp = Float64(1.0 - Float64(1.0 / Float64(2.0 - Float64(Float64(t * Float64(2.0 + Float64(t * -2.0))) * Float64(Float64(Float64(2.0 / t) / Float64(1.0 + Float64(1.0 / t))) - 2.0)))));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -2.6)
		tmp = 0.8333333333333334;
	elseif (t <= 0.8)
		tmp = 1.0 - (1.0 / (2.0 - ((t * (2.0 + (t * -2.0))) * (((2.0 / t) / (1.0 + (1.0 / t))) - 2.0))));
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -2.6], 0.8333333333333334, If[LessEqual[t, 0.8], N[(1.0 - N[(1.0 / N[(2.0 - N[(N[(t * N[(2.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.60000000000000009

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -2.60000000000000009 < 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 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.8:\\ \;\;\;\;1 - \frac{1}{2 - \left(t \cdot \left(2 + t \cdot -2\right)\right) \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(2 + t \cdot -2\right)\\ \mathbf{if}\;t \leq -2.7:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{2 + t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* t (+ 2.0 (* t -2.0)))))
   (if (<= t -2.7)
     0.8333333333333334
     (if (<= t 0.75)
       (+ 1.0 (/ -1.0 (+ 2.0 (* t_1 t_1))))
       (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t)))))))

C

double code(double t) {
	double t_1 = t * (2.0 + (t * -2.0));
	double tmp;
	if (t <= -2.7) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.75) {
		tmp = 1.0 + (-1.0 / (2.0 + (t_1 * t_1)));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (2.0d0 + (t * (-2.0d0)))
    if (t <= (-2.7d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.75d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t_1 * t_1)))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = t * (2.0 + (t * -2.0));
	double tmp;
	if (t <= -2.7) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.75) {
		tmp = 1.0 + (-1.0 / (2.0 + (t_1 * t_1)));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	t_1 = t * (2.0 + (t * -2.0))
	tmp = 0
	if t <= -2.7:
		tmp = 0.8333333333333334
	elif t <= 0.75:
		tmp = 1.0 + (-1.0 / (2.0 + (t_1 * t_1)))
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	t_1 = Float64(t * Float64(2.0 + Float64(t * -2.0)))
	tmp = 0.0
	if (t <= -2.7)
		tmp = 0.8333333333333334;
	elseif (t <= 0.75)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t_1 * t_1))));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = t * (2.0 + (t * -2.0));
	tmp = 0.0;
	if (t <= -2.7)
		tmp = 0.8333333333333334;
	elseif (t <= 0.75)
		tmp = 1.0 + (-1.0 / (2.0 + (t_1 * t_1)));
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(t * N[(2.0 + N[(t * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.7], 0.8333333333333334, If[LessEqual[t, 0.75], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7000000000000002

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -2.7000000000000002 < t < 0.75

   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 0 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in t around 0 99.8%

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

      1. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   if 0.75 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.75:\\ \;\;\;\;1 + \frac{-1}{2 + \left(t \cdot \left(2 + t \cdot -2\right)\right) \cdot \left(t \cdot \left(2 + t \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{-4}{-1 - t} - 4\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -2.2)
   0.8333333333333334
   (if (<= t 0.86)
     (+ 1.0 (/ 1.0 (- (* t (- (/ -4.0 (- -1.0 t)) 4.0)) 2.0)))
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -2.2) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.86) {
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.2d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.86d0) then
        tmp = 1.0d0 + (1.0d0 / ((t * (((-4.0d0) / ((-1.0d0) - t)) - 4.0d0)) - 2.0d0))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -2.2) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.86) {
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -2.2:
		tmp = 0.8333333333333334
	elif t <= 0.86:
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0))
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -2.2)
		tmp = 0.8333333333333334;
	elseif (t <= 0.86)
		tmp = Float64(1.0 + Float64(1.0 / Float64(Float64(t * Float64(Float64(-4.0 / Float64(-1.0 - t)) - 4.0)) - 2.0)));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -2.2)
		tmp = 0.8333333333333334;
	elseif (t <= 0.86)
		tmp = 1.0 + (1.0 / ((t * ((-4.0 / (-1.0 - t)) - 4.0)) - 2.0));
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -2.2], 0.8333333333333334, If[LessEqual[t, 0.86], N[(1.0 + N[(1.0 / N[(N[(t * N[(N[(-4.0 / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.2000000000000002

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -2.2000000000000002 < t < 0.859999999999999987

   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. Step-by-step derivation

      1. *-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. distribute-neg-frac 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l* 99.4%

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

      4. distribute-rgt-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-/l/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in t around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if 0.859999999999999987 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.86:\\ \;\;\;\;1 + \frac{1}{t \cdot \left(\frac{-4}{-1 - t} - 4\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -3.2)
   0.8333333333333334
   (if (<= t 0.7)
     (+ 1.0 (/ -1.0 (+ 2.0 (* t (* t 4.0)))))
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -3.2) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.7) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.2d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.7d0) then
        tmp = 1.0d0 + ((-1.0d0) / (2.0d0 + (t * (t * 4.0d0))))
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -3.2) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.7) {
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -3.2:
		tmp = 0.8333333333333334
	elif t <= 0.7:
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))))
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -3.2)
		tmp = 0.8333333333333334;
	elseif (t <= 0.7)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(2.0 + Float64(t * Float64(t * 4.0)))));
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -3.2)
		tmp = 0.8333333333333334;
	elseif (t <= 0.7)
		tmp = 1.0 + (-1.0 / (2.0 + (t * (t * 4.0))));
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -3.2], 0.8333333333333334, If[LessEqual[t, 0.7], N[(1.0 + N[(-1.0 / N[(2.0 + N[(t * N[(t * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.2000000000000002

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -3.2000000000000002 < t < 0.69999999999999996

   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. Step-by-step derivation

      1. *-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. distribute-neg-frac 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l* 99.4%

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

      4. distribute-rgt-in 99.4%

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

      5. metadata-eval 99.4%

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

      6. associate-/l/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in t around 0 99.3%

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

      1. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   if 0.69999999999999996 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;1 + \frac{-1}{2 + t \cdot \left(t \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34)
   0.8333333333333334
   (if (<= t 0.65)
     0.5
     (- 1.0 (+ 0.16666666666666666 (/ 0.2222222222222222 t))))))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.65d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 - (0.16666666666666666d0 + (0.2222222222222222d0 / t))
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t))
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = Float64(1.0 - Float64(0.16666666666666666 + Float64(0.2222222222222222 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 1.0 - (0.16666666666666666 + (0.2222222222222222 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 0.65], 0.5, N[(1.0 - N[(0.16666666666666666 + N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.340000000000000024

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -0.340000000000000024 < t < 0.650000000000000022

   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 0 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.5 + {t}^{2} \cdot \left(t \cdot \left(2 + -1 \cdot t\right) - 1\right)\right)} \]

   4. Taylor expanded in t around 0 98.9%

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

   if 0.650000000000000022 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \left(0.16666666666666666 + \frac{0.2222222222222222}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34)
   0.8333333333333334
   (if (<= t 0.65) 0.5 (+ 0.8333333333333334 (/ -0.2222222222222222 t)))))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		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 <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 0.65d0) 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 <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 0.65) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 0.65:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t)
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		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 <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 0.65)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334 + (-0.2222222222222222 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 0.65], 0.5, N[(0.8333333333333334 + N[(-0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.340000000000000024

   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%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

   if -0.340000000000000024 < t < 0.650000000000000022

   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 0 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.5 + {t}^{2} \cdot \left(t \cdot \left(2 + -1 \cdot t\right) - 1\right)\right)} \]

   4. Taylor expanded in t around 0 98.9%

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

   if 0.650000000000000022 < 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 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 100.0%

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

      1. cancel-sign-sub-inv 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.65:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 + \frac{-0.2222222222222222}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < -0.340000000000000024 or 1 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around inf 99.3%

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

   if -0.340000000000000024 < 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 0 100.0%

      \[\leadsto 1 - \color{blue}{\left(0.5 + {t}^{2} \cdot \left(t \cdot \left(2 + -1 \cdot t\right) - 1\right)\right)} \]

   4. Taylor expanded in t around 0 98.9%

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

4. Final simplification 99.1%

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

Alternative 9: 58.8% 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%

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

3. Taylor expanded in t around 0 52.5%

   \[\leadsto 1 - \color{blue}{\left(0.5 + {t}^{2} \cdot \left(t \cdot \left(2 + -1 \cdot t\right) - 1\right)\right)} \]

4. Taylor expanded in t around 0 59.9%

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

5. Final simplification 59.9%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Reproduce

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