math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 14

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

Alternative 2: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ t_1 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.0305:\\ \;\;\;\;\cos re + im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im))))
        (t_1 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im -3.1e+154)
     t_0
     (if (<= im -11.5)
       t_1
       (if (<= im 0.0305)
         (+ (cos re) (* im (* (cos re) (* 0.5 im))))
         (if (<= im 2.15e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double t_1 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= -3.1e+154) {
		tmp = t_0;
	} else if (im <= -11.5) {
		tmp = t_1;
	} else if (im <= 0.0305) {
		tmp = cos(re) + (im * (cos(re) * (0.5 * im)));
	} else if (im <= 2.15e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    t_1 = 0.5d0 * (exp(-im) + exp(im))
    if (im <= (-3.1d+154)) then
        tmp = t_0
    else if (im <= (-11.5d0)) then
        tmp = t_1
    else if (im <= 0.0305d0) then
        tmp = cos(re) + (im * (cos(re) * (0.5d0 * im)))
    else if (im <= 2.15d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double t_1 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= -3.1e+154) {
		tmp = t_0;
	} else if (im <= -11.5) {
		tmp = t_1;
	} else if (im <= 0.0305) {
		tmp = Math.cos(re) + (im * (Math.cos(re) * (0.5 * im)));
	} else if (im <= 2.15e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	t_1 = 0.5 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= -3.1e+154:
		tmp = t_0
	elif im <= -11.5:
		tmp = t_1
	elif im <= 0.0305:
		tmp = math.cos(re) + (im * (math.cos(re) * (0.5 * im)))
	elif im <= 2.15e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	t_1 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= -3.1e+154)
		tmp = t_0;
	elseif (im <= -11.5)
		tmp = t_1;
	elseif (im <= 0.0305)
		tmp = Float64(cos(re) + Float64(im * Float64(cos(re) * Float64(0.5 * im))));
	elseif (im <= 2.15e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	t_1 = 0.5 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= -3.1e+154)
		tmp = t_0;
	elseif (im <= -11.5)
		tmp = t_1;
	elseif (im <= 0.0305)
		tmp = cos(re) + (im * (cos(re) * (0.5 * im)));
	elseif (im <= 2.15e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.1e+154], t$95$0, If[LessEqual[im, -11.5], t$95$1, If[LessEqual[im, 0.0305], N[(N[Cos[re], $MachinePrecision] + N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.15e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.1000000000000001e154 or 2.1499999999999999e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto 0.5 \cdot \left(\cos re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      2. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\cos re \cdot im\right) \cdot im\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\cos re \cdot im\right)\right) \cdot im} \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)} \]

      5. *-commutative 100.0%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(im \cdot \cos re\right)\right)} \]

   if -3.1000000000000001e154 < im < -11.5 or 0.030499999999999999 < im < 2.1499999999999999e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right)} \]

   if -11.5 < im < 0.030499999999999999

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 99.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around inf 99.4%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\cos re \cdot {im}^{2}}, \cos re\right) \]
   6. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{{im}^{2} \cdot \cos re}, \cos re\right) \]

      2. unpow2 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(im \cdot im\right)} \cdot \cos re, \cos re\right) \]

      3. associate-*l* 99.4%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot \left(im \cdot \cos re\right)}, \cos re\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot \left(im \cdot \cos re\right)}, \cos re\right) \]
   8. Step-by-step derivation

      1. fma-udef 99.4%

         \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(im \cdot \cos re\right)\right) + \cos re} \]

      2. associate-*r* 99.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \cos re\right)} + \cos re \]

      3. associate-*r* 99.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \cos re} + \cos re \]

      4. *-commutative 99.4%

         \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot 0.5\right)} \cdot \cos re + \cos re \]

      5. associate-*l* 99.4%

         \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot 0.5\right)\right)} \cdot \cos re + \cos re \]

      6. associate-*l* 99.4%

         \[\leadsto \color{blue}{im \cdot \left(\left(im \cdot 0.5\right) \cdot \cos re\right)} + \cos re \]

   9. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(im \cdot 0.5\right) \cdot \cos re\right) + \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0305:\\ \;\;\;\;\cos re + im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ t_1 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq -2.06 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.0071:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im))))
        (t_1 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im -2.06e+154)
     t_0
     (if (<= im -11.5)
       t_1
       (if (<= im 0.0071)
         (* (cos re) (+ 1.0 (* 0.5 (* im im))))
         (if (<= im 1.95e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double t_1 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= -2.06e+154) {
		tmp = t_0;
	} else if (im <= -11.5) {
		tmp = t_1;
	} else if (im <= 0.0071) {
		tmp = cos(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.95e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    t_1 = 0.5d0 * (exp(-im) + exp(im))
    if (im <= (-2.06d+154)) then
        tmp = t_0
    else if (im <= (-11.5d0)) then
        tmp = t_1
    else if (im <= 0.0071d0) then
        tmp = cos(re) * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 1.95d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double t_1 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= -2.06e+154) {
		tmp = t_0;
	} else if (im <= -11.5) {
		tmp = t_1;
	} else if (im <= 0.0071) {
		tmp = Math.cos(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.95e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	t_1 = 0.5 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= -2.06e+154:
		tmp = t_0
	elif im <= -11.5:
		tmp = t_1
	elif im <= 0.0071:
		tmp = math.cos(re) * (1.0 + (0.5 * (im * im)))
	elif im <= 1.95e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	t_1 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= -2.06e+154)
		tmp = t_0;
	elseif (im <= -11.5)
		tmp = t_1;
	elseif (im <= 0.0071)
		tmp = Float64(cos(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.95e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	t_1 = 0.5 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= -2.06e+154)
		tmp = t_0;
	elseif (im <= -11.5)
		tmp = t_1;
	elseif (im <= 0.0071)
		tmp = cos(re) * (1.0 + (0.5 * (im * im)));
	elseif (im <= 1.95e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.06e+154], t$95$0, If[LessEqual[im, -11.5], t$95$1, If[LessEqual[im, 0.0071], N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.95e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.05999999999999988e154 or 1.9500000000000001e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto 0.5 \cdot \left(\cos re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      2. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\cos re \cdot im\right) \cdot im\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\cos re \cdot im\right)\right) \cdot im} \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)} \]

      5. *-commutative 100.0%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(im \cdot \cos re\right)\right)} \]

   if -2.05999999999999988e154 < im < -11.5 or 0.0071000000000000004 < im < 1.9500000000000001e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} + e^{-im}\right)} \]

   if -11.5 < im < 0.0071000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 99.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around inf 99.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]
   6. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{2} \cdot \cos re\right)} + \cos re \]

      2. associate-*l* 99.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \cos re} + \cos re \]

      3. unpow2 99.4%

         \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re + \cos re \]

      4. distribute-lft1-in 99.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re} \]

      5. +-commutative 99.3%

         \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \cos re \]

      6. unpow2 99.3%

         \[\leadsto \left(1 + 0.5 \cdot \color{blue}{{im}^{2}}\right) \cdot \cos re \]

      7. *-commutative 99.3%

         \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot 0.5}\right) \cdot \cos re \]

      8. unpow2 99.3%

         \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \cdot \cos re \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.06 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -11.5:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0071:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 78.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ t_1 := 0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;im \leq -2.9 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+135}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im)))) (t_1 (+ 0.25 (* 0.25 (* re re)))))
   (if (<= im -2.9e+150)
     t_0
     (if (<= im -600.0)
       t_1
       (if (<= im 2.3)
         (cos re)
         (if (<= im 5.4e+135)
           (* (* im im) (+ 0.5 (* (* re re) -0.25)))
           (if (<= im 2.6e+138) t_1 t_0)))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double t_1 = 0.25 + (0.25 * (re * re));
	double tmp;
	if (im <= -2.9e+150) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = t_1;
	} else if (im <= 2.3) {
		tmp = cos(re);
	} else if (im <= 5.4e+135) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else if (im <= 2.6e+138) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    t_1 = 0.25d0 + (0.25d0 * (re * re))
    if (im <= (-2.9d+150)) then
        tmp = t_0
    else if (im <= (-600.0d0)) then
        tmp = t_1
    else if (im <= 2.3d0) then
        tmp = cos(re)
    else if (im <= 5.4d+135) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else if (im <= 2.6d+138) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double t_1 = 0.25 + (0.25 * (re * re));
	double tmp;
	if (im <= -2.9e+150) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = t_1;
	} else if (im <= 2.3) {
		tmp = Math.cos(re);
	} else if (im <= 5.4e+135) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else if (im <= 2.6e+138) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	t_1 = 0.25 + (0.25 * (re * re))
	tmp = 0
	if im <= -2.9e+150:
		tmp = t_0
	elif im <= -600.0:
		tmp = t_1
	elif im <= 2.3:
		tmp = math.cos(re)
	elif im <= 5.4e+135:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	elif im <= 2.6e+138:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	t_1 = Float64(0.25 + Float64(0.25 * Float64(re * re)))
	tmp = 0.0
	if (im <= -2.9e+150)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = t_1;
	elseif (im <= 2.3)
		tmp = cos(re);
	elseif (im <= 5.4e+135)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	elseif (im <= 2.6e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	t_1 = 0.25 + (0.25 * (re * re));
	tmp = 0.0;
	if (im <= -2.9e+150)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = t_1;
	elseif (im <= 2.3)
		tmp = cos(re);
	elseif (im <= 5.4e+135)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	elseif (im <= 2.6e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.9e+150], t$95$0, If[LessEqual[im, -600.0], t$95$1, If[LessEqual[im, 2.3], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.4e+135], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+138], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.90000000000000011e150 or 2.6000000000000001e138 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 92.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 92.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 92.5%

         \[\leadsto 0.5 \cdot \left(\cos re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      2. associate-*r* 92.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\cos re \cdot im\right) \cdot im\right)} \]

      3. associate-*r* 92.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\cos re \cdot im\right)\right) \cdot im} \]

      4. *-commutative 92.5%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)} \]

      5. *-commutative 92.5%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   7. Simplified 92.5%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(im \cdot \cos re\right)\right)} \]

   if -2.90000000000000011e150 < im < -600 or 5.3999999999999997e135 < im < 2.6000000000000001e138

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   3. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

   4. Taylor expanded in re around 0 18.7%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 18.7%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

      2. unpow2 18.7%

         \[\leadsto 0.25 + \color{blue}{\left(re \cdot re\right)} \cdot 0.25 \]

   6. Simplified 18.7%

      \[\leadsto \color{blue}{0.25 + \left(re \cdot re\right) \cdot 0.25} \]

   if -600 < im < 2.2999999999999998

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 98.1%

      \[\leadsto \color{blue}{\cos re} \]

   if 2.2999999999999998 < im < 5.3999999999999997e135

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 5.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 5.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 5.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.8%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 5.8%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 5.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 5.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 27.6%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 27.6%

         \[\leadsto 0.5 \cdot {im}^{2} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} \]

      2. distribute-rgt-out 27.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      3. unpow2 27.6%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right) \]

      4. +-commutative 27.6%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      5. *-commutative 27.6%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + 0.5\right) \]

      6. unpow2 27.6%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25 + 0.5\right) \]

   10. Simplified 27.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+135}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+138}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 76.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{if}\;im \leq -4 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -255:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 320000:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im)))))
   (if (<= im -4e+150)
     t_0
     (if (<= im -255.0)
       (+ 0.25 (* 0.25 (* re re)))
       (if (<= im 320000.0) (+ (cos re) (* im (* 0.5 im))) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double tmp;
	if (im <= -4e+150) {
		tmp = t_0;
	} else if (im <= -255.0) {
		tmp = 0.25 + (0.25 * (re * re));
	} else if (im <= 320000.0) {
		tmp = cos(re) + (im * (0.5 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    if (im <= (-4d+150)) then
        tmp = t_0
    else if (im <= (-255.0d0)) then
        tmp = 0.25d0 + (0.25d0 * (re * re))
    else if (im <= 320000.0d0) then
        tmp = cos(re) + (im * (0.5d0 * im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double tmp;
	if (im <= -4e+150) {
		tmp = t_0;
	} else if (im <= -255.0) {
		tmp = 0.25 + (0.25 * (re * re));
	} else if (im <= 320000.0) {
		tmp = Math.cos(re) + (im * (0.5 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	tmp = 0
	if im <= -4e+150:
		tmp = t_0
	elif im <= -255.0:
		tmp = 0.25 + (0.25 * (re * re))
	elif im <= 320000.0:
		tmp = math.cos(re) + (im * (0.5 * im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	tmp = 0.0
	if (im <= -4e+150)
		tmp = t_0;
	elseif (im <= -255.0)
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	elseif (im <= 320000.0)
		tmp = Float64(cos(re) + Float64(im * Float64(0.5 * im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	tmp = 0.0;
	if (im <= -4e+150)
		tmp = t_0;
	elseif (im <= -255.0)
		tmp = 0.25 + (0.25 * (re * re));
	elseif (im <= 320000.0)
		tmp = cos(re) + (im * (0.5 * im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4e+150], t$95$0, If[LessEqual[im, -255.0], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 320000.0], N[(N[Cos[re], $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.99999999999999992e150 or 3.2e5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 73.9%

         \[\leadsto 0.5 \cdot \left(\cos re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      2. associate-*r* 73.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\cos re \cdot im\right) \cdot im\right)} \]

      3. associate-*r* 73.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\cos re \cdot im\right)\right) \cdot im} \]

      4. *-commutative 73.9%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)} \]

      5. *-commutative 73.9%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   7. Simplified 73.9%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(im \cdot \cos re\right)\right)} \]

   if -3.99999999999999992e150 < im < -255

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   3. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

   4. Taylor expanded in re around 0 16.8%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 16.8%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

      2. unpow2 16.8%

         \[\leadsto 0.25 + \color{blue}{\left(re \cdot re\right)} \cdot 0.25 \]

   6. Simplified 16.8%

      \[\leadsto \color{blue}{0.25 + \left(re \cdot re\right) \cdot 0.25} \]

   if -255 < im < 3.2e5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around 0 96.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{{im}^{2}}, \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 96.8%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, \cos re\right) \]

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, \cos re\right) \]
   8. Step-by-step derivation

      1. fma-udef 96.8%

         \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right) + \cos re} \]

      2. *-commutative 96.8%

         \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot 0.5} + \cos re \]

      3. associate-*l* 96.8%

         \[\leadsto \color{blue}{im \cdot \left(im \cdot 0.5\right)} + \cos re \]

   9. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{im \cdot \left(im \cdot 0.5\right) + \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -255:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 320000:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 77.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{if}\;im \leq -6.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot \left(im \cdot 0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 21000000:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im)))))
   (if (<= im -6.6e+150)
     t_0
     (if (<= im -600.0)
       (* (pow re 4.0) (* im (* im 0.020833333333333332)))
       (if (<= im 21000000.0) (+ (cos re) (* im (* 0.5 im))) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double tmp;
	if (im <= -6.6e+150) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = pow(re, 4.0) * (im * (im * 0.020833333333333332));
	} else if (im <= 21000000.0) {
		tmp = cos(re) + (im * (0.5 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    if (im <= (-6.6d+150)) then
        tmp = t_0
    else if (im <= (-600.0d0)) then
        tmp = (re ** 4.0d0) * (im * (im * 0.020833333333333332d0))
    else if (im <= 21000000.0d0) then
        tmp = cos(re) + (im * (0.5d0 * im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double tmp;
	if (im <= -6.6e+150) {
		tmp = t_0;
	} else if (im <= -600.0) {
		tmp = Math.pow(re, 4.0) * (im * (im * 0.020833333333333332));
	} else if (im <= 21000000.0) {
		tmp = Math.cos(re) + (im * (0.5 * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	tmp = 0
	if im <= -6.6e+150:
		tmp = t_0
	elif im <= -600.0:
		tmp = math.pow(re, 4.0) * (im * (im * 0.020833333333333332))
	elif im <= 21000000.0:
		tmp = math.cos(re) + (im * (0.5 * im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	tmp = 0.0
	if (im <= -6.6e+150)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = Float64((re ^ 4.0) * Float64(im * Float64(im * 0.020833333333333332)));
	elseif (im <= 21000000.0)
		tmp = Float64(cos(re) + Float64(im * Float64(0.5 * im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	tmp = 0.0;
	if (im <= -6.6e+150)
		tmp = t_0;
	elseif (im <= -600.0)
		tmp = (re ^ 4.0) * (im * (im * 0.020833333333333332));
	elseif (im <= 21000000.0)
		tmp = cos(re) + (im * (0.5 * im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.6e+150], t$95$0, If[LessEqual[im, -600.0], N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * N[(im * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 21000000.0], N[(N[Cos[re], $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.59999999999999962e150 or 2.1e7 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 73.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 73.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 73.9%

         \[\leadsto 0.5 \cdot \left(\cos re \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      2. associate-*r* 73.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\cos re \cdot im\right) \cdot im\right)} \]

      3. associate-*r* 73.9%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(\cos re \cdot im\right)\right) \cdot im} \]

      4. *-commutative 73.9%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)} \]

      5. *-commutative 73.9%

         \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   7. Simplified 73.9%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(im \cdot \cos re\right)\right)} \]

   if -6.59999999999999962e150 < im < -600

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 5.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 5.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 5.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 5.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 5.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 5.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 7.6%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + \left(0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 7.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]

      2. +-commutative 7.6%

         \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right) + \left(0.5 \cdot {im}^{2} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right)} \]

      3. associate-*r* 7.6%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} + \left(0.5 \cdot {im}^{2} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) \]

      4. *-commutative 7.6%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right)} + \left(0.5 \cdot {im}^{2} + 0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)\right) \]

      5. associate-*r* 7.6%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + \left(0.5 \cdot {im}^{2} + \color{blue}{\left(0.020833333333333332 \cdot {re}^{4}\right) \cdot {im}^{2}}\right) \]

      6. distribute-rgt-out 7.6%

         \[\leadsto {im}^{2} \cdot \left(-0.25 \cdot {re}^{2}\right) + \color{blue}{{im}^{2} \cdot \left(0.5 + 0.020833333333333332 \cdot {re}^{4}\right)} \]

      7. distribute-lft-out 14.8%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(-0.25 \cdot {re}^{2} + \left(0.5 + 0.020833333333333332 \cdot {re}^{4}\right)\right)} \]

      8. unpow2 14.8%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(-0.25 \cdot {re}^{2} + \left(0.5 + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      9. *-commutative 14.8%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + \left(0.5 + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      10. unpow2 14.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25 + \left(0.5 + 0.020833333333333332 \cdot {re}^{4}\right)\right) \]

      11. *-commutative 14.8%

          \[\leadsto \left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + \left(0.5 + \color{blue}{{re}^{4} \cdot 0.020833333333333332}\right)\right) \]

   10. Simplified 14.8%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + \left(0.5 + {re}^{4} \cdot 0.020833333333333332\right)\right)} \]

   11. Taylor expanded in re around inf 27.2%

       \[\leadsto \color{blue}{0.020833333333333332 \cdot \left({re}^{4} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 27.2%

          \[\leadsto \color{blue}{\left({re}^{4} \cdot {im}^{2}\right) \cdot 0.020833333333333332} \]

       2. associate-*r* 27.2%

          \[\leadsto \color{blue}{{re}^{4} \cdot \left({im}^{2} \cdot 0.020833333333333332\right)} \]

       3. unpow2 27.2%

          \[\leadsto {re}^{4} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.020833333333333332\right) \]

       4. associate-*l* 27.2%

          \[\leadsto {re}^{4} \cdot \color{blue}{\left(im \cdot \left(im \cdot 0.020833333333333332\right)\right)} \]

   13. Simplified 27.2%

       \[\leadsto \color{blue}{{re}^{4} \cdot \left(im \cdot \left(im \cdot 0.020833333333333332\right)\right)} \]

   if -600 < im < 2.1e7

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around 0 96.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{{im}^{2}}, \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 96.8%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, \cos re\right) \]

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, \cos re\right) \]
   8. Step-by-step derivation

      1. fma-udef 96.8%

         \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot im\right) + \cos re} \]

      2. *-commutative 96.8%

         \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot 0.5} + \cos re \]

      3. associate-*l* 96.8%

         \[\leadsto \color{blue}{im \cdot \left(im \cdot 0.5\right)} + \cos re \]

   9. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{im \cdot \left(im \cdot 0.5\right) + \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.6 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -600:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot \left(im \cdot 0.020833333333333332\right)\right)\\ \mathbf{elif}\;im \leq 21000000:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (cos re) (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	return cos(re) * (1.0 + (0.5 * (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * (1.0d0 + (0.5d0 * (im * im)))
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * (1.0 + (0.5 * (im * im)));
}

Python

def code(re, im):
	return math.cos(re) * (1.0 + (0.5 * (im * im)))

Julia

function code(re, im)
	return Float64(cos(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * (1.0 + (0.5 * (im * im)));
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Taylor expanded in im around 0 75.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

4. Simplified 75.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

5. Taylor expanded in re around inf 75.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]
6. Step-by-step derivation

   1. *-commutative 75.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{2} \cdot \cos re\right)} + \cos re \]

   2. associate-*l* 75.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2}\right) \cdot \cos re} + \cos re \]

   3. unpow2 75.1%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot \cos re + \cos re \]

   4. distribute-lft1-in 75.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot \left(im \cdot im\right) + 1\right) \cdot \cos re} \]

   5. +-commutative 75.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \cdot \cos re \]

   6. unpow2 75.1%

      \[\leadsto \left(1 + 0.5 \cdot \color{blue}{{im}^{2}}\right) \cdot \cos re \]

   7. *-commutative 75.1%

      \[\leadsto \left(1 + \color{blue}{{im}^{2} \cdot 0.5}\right) \cdot \cos re \]

   8. unpow2 75.1%

      \[\leadsto \left(1 + \color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \cdot \cos re \]

7. Simplified 75.1%

   \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot 0.5\right) \cdot \cos re} \]

8. Final simplification 75.1%

   \[\leadsto \cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

Alternative 8: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{+23} \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.8e+23) (not (<= im 2.5)))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.8e+23) || !(im <= 2.5)) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.8d+23)) .or. (.not. (im <= 2.5d0))) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.8e+23) || !(im <= 2.5)) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.8e+23) or not (im <= 2.5):
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.8e+23) || !(im <= 2.5))
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.8e+23) || ~((im <= 2.5)))
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.8e+23], N[Not[LessEqual[im, 2.5]], $MachinePrecision]], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.7999999999999999e23 or 2.5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 51.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 51.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 51.4%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 51.4%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 51.4%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 51.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 10.0%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 10.0%

         \[\leadsto 0.5 \cdot {im}^{2} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} \]

      2. distribute-rgt-out 49.3%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      3. unpow2 49.3%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right) \]

      4. +-commutative 49.3%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      5. *-commutative 49.3%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + 0.5\right) \]

      6. unpow2 49.3%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25 + 0.5\right) \]

   10. Simplified 49.3%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]

   if -1.7999999999999999e23 < im < 2.5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 93.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.8 \cdot 10^{+23} \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \]

Alternative 9: 51.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 320000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{im}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (+ 0.5 (* (* re re) -0.25)))))
   (if (<= im -6.8e+23)
     t_0
     (if (<= im 320000.0)
       (+ 1.0 (* 0.5 (* im im)))
       (if (<= im 1e+161) t_0 (* im (/ im 2.0)))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -6.8e+23) {
		tmp = t_0;
	} else if (im <= 320000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 1e+161) {
		tmp = t_0;
	} else {
		tmp = im * (im / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    if (im <= (-6.8d+23)) then
        tmp = t_0
    else if (im <= 320000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (im * im))
    else if (im <= 1d+161) then
        tmp = t_0
    else
        tmp = im * (im / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	double tmp;
	if (im <= -6.8e+23) {
		tmp = t_0;
	} else if (im <= 320000.0) {
		tmp = 1.0 + (0.5 * (im * im));
	} else if (im <= 1e+161) {
		tmp = t_0;
	} else {
		tmp = im * (im / 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (0.5 + ((re * re) * -0.25))
	tmp = 0
	if im <= -6.8e+23:
		tmp = t_0
	elif im <= 320000.0:
		tmp = 1.0 + (0.5 * (im * im))
	elif im <= 1e+161:
		tmp = t_0
	else:
		tmp = im * (im / 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)))
	tmp = 0.0
	if (im <= -6.8e+23)
		tmp = t_0;
	elseif (im <= 320000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	elseif (im <= 1e+161)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(im / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (0.5 + ((re * re) * -0.25));
	tmp = 0.0;
	if (im <= -6.8e+23)
		tmp = t_0;
	elseif (im <= 320000.0)
		tmp = 1.0 + (0.5 * (im * im));
	elseif (im <= 1e+161)
		tmp = t_0;
	else
		tmp = im * (im / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.8e+23], t$95$0, If[LessEqual[im, 320000.0], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+161], t$95$0, N[(im * N[(im / 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.79999999999999983e23 or 3.2e5 < im < 1e161

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 40.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 40.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 40.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 40.8%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 40.8%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 40.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 40.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 12.4%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 12.4%

         \[\leadsto 0.5 \cdot {im}^{2} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} \]

      2. distribute-rgt-out 45.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      3. unpow2 45.0%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right) \]

      4. +-commutative 45.0%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      5. *-commutative 45.0%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + 0.5\right) \]

      6. unpow2 45.0%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25 + 0.5\right) \]

   10. Simplified 45.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]

   if -6.79999999999999983e23 < im < 3.2e5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 92.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around 0 54.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 54.5%

         \[\leadsto 1 + \color{blue}{{im}^{2} \cdot 0.5} \]

      2. unpow2 54.5%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot 0.5 \]

   7. Simplified 54.5%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot 0.5} \]

   if 1e161 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 85.7%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 85.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. rem-log-exp 85.7%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\log \left(e^{im}\right)}\right) \]

      3. log-pow 85.7%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left({\left(e^{im}\right)}^{im}\right)} \]

      4. log-pow 85.7%

         \[\leadsto \color{blue}{\log \left({\left({\left(e^{im}\right)}^{im}\right)}^{0.5}\right)} \]

      5. unpow1/2 85.7%

         \[\leadsto \log \color{blue}{\left(\sqrt{{\left(e^{im}\right)}^{im}}\right)} \]

      6. sqr-pow 85.7%

         \[\leadsto \log \left(\sqrt{\color{blue}{{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}}}\right) \]

      7. rem-sqrt-square 85.7%

         \[\leadsto \log \color{blue}{\left(\left|{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right|\right)} \]

      8. sqr-pow 85.7%

         \[\leadsto \log \left(\left|\color{blue}{{\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}}\right|\right) \]

      9. fabs-sqr 85.7%

         \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}\right)} \]

      10. sqr-pow 85.7%

          \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right)} \]

      11. log-pow 85.7%

          \[\leadsto \color{blue}{\frac{im}{2} \cdot \log \left(e^{im}\right)} \]

      12. rem-log-exp 85.7%

          \[\leadsto \frac{im}{2} \cdot \color{blue}{im} \]

   10. Simplified 85.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.8 \cdot 10^{+23}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 320000:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+161}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{im}{2}\\ \end{array} \]

Alternative 10: 28.9% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \frac{im}{2}\\ t_1 := 0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;im \leq -6.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -410:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (/ im 2.0))) (t_1 (+ 0.25 (* 0.25 (* re re)))))
   (if (<= im -6.6e+150)
     t_0
     (if (<= im -410.0)
       t_1
       (if (<= im 1.3e+15) 0.25 (if (<= im 1.8e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (im / 2.0);
	double t_1 = 0.25 + (0.25 * (re * re));
	double tmp;
	if (im <= -6.6e+150) {
		tmp = t_0;
	} else if (im <= -410.0) {
		tmp = t_1;
	} else if (im <= 1.3e+15) {
		tmp = 0.25;
	} else if (im <= 1.8e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (im / 2.0d0)
    t_1 = 0.25d0 + (0.25d0 * (re * re))
    if (im <= (-6.6d+150)) then
        tmp = t_0
    else if (im <= (-410.0d0)) then
        tmp = t_1
    else if (im <= 1.3d+15) then
        tmp = 0.25d0
    else if (im <= 1.8d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (im / 2.0);
	double t_1 = 0.25 + (0.25 * (re * re));
	double tmp;
	if (im <= -6.6e+150) {
		tmp = t_0;
	} else if (im <= -410.0) {
		tmp = t_1;
	} else if (im <= 1.3e+15) {
		tmp = 0.25;
	} else if (im <= 1.8e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (im / 2.0)
	t_1 = 0.25 + (0.25 * (re * re))
	tmp = 0
	if im <= -6.6e+150:
		tmp = t_0
	elif im <= -410.0:
		tmp = t_1
	elif im <= 1.3e+15:
		tmp = 0.25
	elif im <= 1.8e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(im / 2.0))
	t_1 = Float64(0.25 + Float64(0.25 * Float64(re * re)))
	tmp = 0.0
	if (im <= -6.6e+150)
		tmp = t_0;
	elseif (im <= -410.0)
		tmp = t_1;
	elseif (im <= 1.3e+15)
		tmp = 0.25;
	elseif (im <= 1.8e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (im / 2.0);
	t_1 = 0.25 + (0.25 * (re * re));
	tmp = 0.0;
	if (im <= -6.6e+150)
		tmp = t_0;
	elseif (im <= -410.0)
		tmp = t_1;
	elseif (im <= 1.3e+15)
		tmp = 0.25;
	elseif (im <= 1.8e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(im / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.6e+150], t$95$0, If[LessEqual[im, -410.0], t$95$1, If[LessEqual[im, 1.3e+15], 0.25, If[LessEqual[im, 1.8e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.59999999999999962e150 or 1.8e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 98.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 98.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 98.4%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 98.4%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 98.4%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 63.8%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 63.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. rem-log-exp 65.5%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\log \left(e^{im}\right)}\right) \]

      3. log-pow 65.5%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left({\left(e^{im}\right)}^{im}\right)} \]

      4. log-pow 65.5%

         \[\leadsto \color{blue}{\log \left({\left({\left(e^{im}\right)}^{im}\right)}^{0.5}\right)} \]

      5. unpow1/2 65.5%

         \[\leadsto \log \color{blue}{\left(\sqrt{{\left(e^{im}\right)}^{im}}\right)} \]

      6. sqr-pow 65.5%

         \[\leadsto \log \left(\sqrt{\color{blue}{{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}}}\right) \]

      7. rem-sqrt-square 65.5%

         \[\leadsto \log \color{blue}{\left(\left|{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right|\right)} \]

      8. sqr-pow 65.5%

         \[\leadsto \log \left(\left|\color{blue}{{\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}}\right|\right) \]

      9. fabs-sqr 65.5%

         \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}\right)} \]

      10. sqr-pow 65.5%

          \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right)} \]

      11. log-pow 65.5%

          \[\leadsto \color{blue}{\frac{im}{2} \cdot \log \left(e^{im}\right)} \]

      12. rem-log-exp 63.8%

          \[\leadsto \frac{im}{2} \cdot \color{blue}{im} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\frac{im}{2} \cdot im} \]

   if -6.59999999999999962e150 < im < -410 or 1.3e15 < im < 1.8e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   3. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

   4. Taylor expanded in re around 0 17.3%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 17.3%

         \[\leadsto 0.25 + \color{blue}{{re}^{2} \cdot 0.25} \]

      2. unpow2 17.3%

         \[\leadsto 0.25 + \color{blue}{\left(re \cdot re\right)} \cdot 0.25 \]

   6. Simplified 17.3%

      \[\leadsto \color{blue}{0.25 + \left(re \cdot re\right) \cdot 0.25} \]

   if -410 < im < 1.3e15

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   3. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

   4. Taylor expanded in re around 0 13.8%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.6 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \frac{im}{2}\\ \mathbf{elif}\;im \leq -410:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{im}{2}\\ \end{array} \]

Alternative 11: 49.7% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+167} \lor \neg \left(re \leq 1.56\right):\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -4.5e+167) (not (<= re 1.56)))
   (* im (* (* re re) (* im -0.25)))
   (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -4.5e+167) || !(re <= 1.56)) {
		tmp = im * ((re * re) * (im * -0.25));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-4.5d+167)) .or. (.not. (re <= 1.56d0))) then
        tmp = im * ((re * re) * (im * (-0.25d0)))
    else
        tmp = 1.0d0 + (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -4.5e+167) || !(re <= 1.56)) {
		tmp = im * ((re * re) * (im * -0.25));
	} else {
		tmp = 1.0 + (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -4.5e+167) or not (re <= 1.56):
		tmp = im * ((re * re) * (im * -0.25))
	else:
		tmp = 1.0 + (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -4.5e+167) || !(re <= 1.56))
		tmp = Float64(im * Float64(Float64(re * re) * Float64(im * -0.25)));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -4.5e+167) || ~((re <= 1.56)))
		tmp = im * ((re * re) * (im * -0.25));
	else
		tmp = 1.0 + (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -4.5e+167], N[Not[LessEqual[re, 1.56]], $MachinePrecision]], N[(im * N[(N[(re * re), $MachinePrecision] * N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.4999999999999999e167 or 1.5600000000000001 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 76.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 29.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 29.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 29.7%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 29.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 29.7%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 11.4%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2} + -0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 11.4%

         \[\leadsto 0.5 \cdot {im}^{2} + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot {im}^{2}} \]

      2. distribute-rgt-out 29.5%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      3. unpow2 29.5%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right) \]

      4. +-commutative 29.5%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      5. *-commutative 29.5%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{{re}^{2} \cdot -0.25} + 0.5\right) \]

      6. unpow2 29.5%

         \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25 + 0.5\right) \]

   10. Simplified 29.5%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]

   11. Taylor expanded in re around inf 29.5%

       \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot {im}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 29.5%

          \[\leadsto \color{blue}{\left({re}^{2} \cdot {im}^{2}\right) \cdot -0.25} \]

       2. *-commutative 29.5%

          \[\leadsto \color{blue}{\left({im}^{2} \cdot {re}^{2}\right)} \cdot -0.25 \]

       3. associate-*r* 29.5%

          \[\leadsto \color{blue}{{im}^{2} \cdot \left({re}^{2} \cdot -0.25\right)} \]

       4. unpow2 29.5%

          \[\leadsto {im}^{2} \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

       5. associate-*r* 29.5%

          \[\leadsto {im}^{2} \cdot \color{blue}{\left(re \cdot \left(re \cdot -0.25\right)\right)} \]

       6. unpow2 29.5%

          \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(re \cdot \left(re \cdot -0.25\right)\right) \]

       7. associate-*l* 29.7%

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(re \cdot \left(re \cdot -0.25\right)\right)\right)} \]

       8. associate-*r* 29.7%

          \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot -0.25\right)}\right) \]

       9. unpow2 29.7%

          \[\leadsto im \cdot \left(im \cdot \left(\color{blue}{{re}^{2}} \cdot -0.25\right)\right) \]

       10. associate-*l* 29.7%

           \[\leadsto im \cdot \color{blue}{\left(\left(im \cdot {re}^{2}\right) \cdot -0.25\right)} \]

       11. *-commutative 29.7%

           \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot im\right)} \cdot -0.25\right) \]

       12. associate-*l* 29.7%

           \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \left(im \cdot -0.25\right)\right)} \]

       13. unpow2 29.7%

           \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot -0.25\right)\right) \]

   13. Simplified 29.7%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot -0.25\right)\right)} \]

   if -4.4999999999999999e167 < re < 1.5600000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 74.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in re around 0 62.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto 1 + \color{blue}{{im}^{2} \cdot 0.5} \]

      2. unpow2 62.5%

         \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot 0.5 \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+167} \lor \neg \left(re \leq 1.56\right):\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 12: 26.3% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -11.5 \lor \neg \left(im \leq 0.7\right):\\ \;\;\;\;im \cdot \frac{im}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -11.5) (not (<= im 0.7))) (* im (/ im 2.0)) 0.25))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -11.5) || !(im <= 0.7)) {
		tmp = im * (im / 2.0);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-11.5d0)) .or. (.not. (im <= 0.7d0))) then
        tmp = im * (im / 2.0d0)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -11.5) || !(im <= 0.7)) {
		tmp = im * (im / 2.0);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -11.5) or not (im <= 0.7):
		tmp = im * (im / 2.0)
	else:
		tmp = 0.25
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -11.5) || !(im <= 0.7))
		tmp = Float64(im * Float64(im / 2.0));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -11.5) || ~((im <= 0.7)))
		tmp = im * (im / 2.0);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -11.5], N[Not[LessEqual[im, 0.7]], $MachinePrecision]], N[(im * N[(im / 2.0), $MachinePrecision]), $MachinePrecision], 0.25]

Derivation

1. Split input into 2 regimes

2. if im < -11.5 or 0.69999999999999996 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

   4. Simplified 48.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

   5. Taylor expanded in im around inf 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 48.2%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*r* 48.2%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. unpow2 48.2%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot 0.5\right) \]

   7. Simplified 48.2%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(im \cdot im\right) \cdot 0.5\right)} \]

   8. Taylor expanded in re around 0 32.0%

      \[\leadsto \color{blue}{0.5 \cdot {im}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 32.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot im\right)} \]

      2. rem-log-exp 73.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\log \left(e^{im}\right)}\right) \]

      3. log-pow 73.5%

         \[\leadsto 0.5 \cdot \color{blue}{\log \left({\left(e^{im}\right)}^{im}\right)} \]

      4. log-pow 73.5%

         \[\leadsto \color{blue}{\log \left({\left({\left(e^{im}\right)}^{im}\right)}^{0.5}\right)} \]

      5. unpow1/2 73.5%

         \[\leadsto \log \color{blue}{\left(\sqrt{{\left(e^{im}\right)}^{im}}\right)} \]

      6. sqr-pow 73.5%

         \[\leadsto \log \left(\sqrt{\color{blue}{{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}}}\right) \]

      7. rem-sqrt-square 73.5%

         \[\leadsto \log \color{blue}{\left(\left|{\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right|\right)} \]

      8. sqr-pow 73.5%

         \[\leadsto \log \left(\left|\color{blue}{{\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}}\right|\right) \]

      9. fabs-sqr 73.5%

         \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)} \cdot {\left(e^{im}\right)}^{\left(\frac{\frac{im}{2}}{2}\right)}\right)} \]

      10. sqr-pow 73.5%

          \[\leadsto \log \color{blue}{\left({\left(e^{im}\right)}^{\left(\frac{im}{2}\right)}\right)} \]

      11. log-pow 73.6%

          \[\leadsto \color{blue}{\frac{im}{2} \cdot \log \left(e^{im}\right)} \]

      12. rem-log-exp 32.0%

          \[\leadsto \frac{im}{2} \cdot \color{blue}{im} \]

   10. Simplified 32.0%

       \[\leadsto \color{blue}{\frac{im}{2} \cdot im} \]

   if -11.5 < im < 0.69999999999999996

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   3. Applied egg-rr 13.7%

      \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

   4. Taylor expanded in re around 0 14.0%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -11.5 \lor \neg \left(im \leq 0.7\right):\\ \;\;\;\;im \cdot \frac{im}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 13: 47.0% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(im \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 (* 0.5 (* im im))))

C

double code(double re, double im) {
	return 1.0 + (0.5 * (im * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + (0.5d0 * (im * im))
end function

Java

public static double code(double re, double im) {
	return 1.0 + (0.5 * (im * im));
}

Python

def code(re, im):
	return 1.0 + (0.5 * (im * im))

Julia

function code(re, im)
	return Float64(1.0 + Float64(0.5 * Float64(im * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + (0.5 * (im * im));
end

Wolfram

code[re_, im_] := N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Taylor expanded in im around 0 75.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot {im}^{2}\right) + \cos re} \]

4. Simplified 75.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \cos re \cdot \left(im \cdot im\right), \cos re\right)} \]

5. Taylor expanded in re around 0 45.9%

   \[\leadsto \color{blue}{1 + 0.5 \cdot {im}^{2}} \]
6. Step-by-step derivation

   1. *-commutative 45.9%

      \[\leadsto 1 + \color{blue}{{im}^{2} \cdot 0.5} \]

   2. unpow2 45.9%

      \[\leadsto 1 + \color{blue}{\left(im \cdot im\right)} \cdot 0.5 \]

7. Simplified 45.9%

   \[\leadsto \color{blue}{1 + \left(im \cdot im\right) \cdot 0.5} \]

8. Final simplification 45.9%

   \[\leadsto 1 + 0.5 \cdot \left(im \cdot im\right) \]

Alternative 14: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

3. Applied egg-rr 8.5%

   \[\leadsto \color{blue}{{\left(\cos re \cdot -2\right)}^{-2}} \]

4. Taylor expanded in re around 0 8.7%

   \[\leadsto \color{blue}{0.25} \]

5. Final simplification 8.7%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023193 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))