math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 15

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.3% 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: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ t_1 := \left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\cos re + t_0\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im))))
        (t_1 (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))
   (if (<= im -7.2e+150)
     t_0
     (if (<= im -2.6)
       t_1
       (if (<= im 1.95)
         (+ (cos re) t_0)
         (if (<= im 2.1e+150) t_1 (* im (* 0.5 (* (cos re) im)))))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double t_1 = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -7.2e+150) {
		tmp = t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 1.95) {
		tmp = cos(re) + t_0;
	} else if (im <= 2.1e+150) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (cos(re) * im));
	}
	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 = cos(re) * (0.5d0 * (im * im))
    t_1 = (exp(-im) + exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    if (im <= (-7.2d+150)) then
        tmp = t_0
    else if (im <= (-2.6d0)) then
        tmp = t_1
    else if (im <= 1.95d0) then
        tmp = cos(re) + t_0
    else if (im <= 2.1d+150) then
        tmp = t_1
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double t_1 = (Math.exp(-im) + Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	double tmp;
	if (im <= -7.2e+150) {
		tmp = t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 1.95) {
		tmp = Math.cos(re) + t_0;
	} else if (im <= 2.1e+150) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	t_1 = (math.exp(-im) + math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	tmp = 0
	if im <= -7.2e+150:
		tmp = t_0
	elif im <= -2.6:
		tmp = t_1
	elif im <= 1.95:
		tmp = math.cos(re) + t_0
	elif im <= 2.1e+150:
		tmp = t_1
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	t_1 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))))
	tmp = 0.0
	if (im <= -7.2e+150)
		tmp = t_0;
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 1.95)
		tmp = Float64(cos(re) + t_0);
	elseif (im <= 2.1e+150)
		tmp = t_1;
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	t_1 = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	tmp = 0.0;
	if (im <= -7.2e+150)
		tmp = t_0;
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 1.95)
		tmp = cos(re) + t_0;
	elseif (im <= 2.1e+150)
		tmp = t_1;
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+150], t$95$0, If[LessEqual[im, -2.6], t$95$1, If[LessEqual[im, 1.95], N[(N[Cos[re], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[im, 2.1e+150], t$95$1, N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.19999999999999972e150

   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 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in im around inf 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*r* 96.7%

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

      3. unpow2 96.7%

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

   7. Simplified 96.7%

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

   if -7.19999999999999972e150 < im < -2.60000000000000009 or 1.94999999999999996 < im < 2.09999999999999998e150

   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 0.0%

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

   4. Simplified 77.8%

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

   if -2.60000000000000009 < im < 1.94999999999999996

   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 98.8%

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

   4. Simplified 98.8%

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

      1. fma-udef 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 2.09999999999999998e150 < 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 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around inf 96.9%

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

      1. unpow2 96.9%

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

      2. associate-*r* 96.9%

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

      3. associate-*r* 96.9%

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

      4. *-commutative 96.9%

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

      5. *-commutative 96.9%

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

   7. Simplified 96.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+150}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.95:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ t_1 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;\cos re + t_0\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* 0.5 (* im im))))
        (t_1 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im -1.35e+154)
     t_0
     (if (<= im -2.6)
       t_1
       (if (<= im 2000000000.0)
         (+ (cos re) t_0)
         (if (<= im 3.1e+154) t_1 (* im (* 0.5 (* (cos re) im)))))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (0.5 * (im * im));
	double t_1 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 2000000000.0) {
		tmp = cos(re) + t_0;
	} else if (im <= 3.1e+154) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (cos(re) * im));
	}
	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 = cos(re) * (0.5d0 * (im * im))
    t_1 = 0.5d0 * (exp(-im) + exp(im))
    if (im <= (-1.35d+154)) then
        tmp = t_0
    else if (im <= (-2.6d0)) then
        tmp = t_1
    else if (im <= 2000000000.0d0) then
        tmp = cos(re) + t_0
    else if (im <= 3.1d+154) then
        tmp = t_1
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (0.5 * (im * im));
	double t_1 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 2000000000.0) {
		tmp = Math.cos(re) + t_0;
	} else if (im <= 3.1e+154) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (0.5 * (im * im))
	t_1 = 0.5 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_0
	elif im <= -2.6:
		tmp = t_1
	elif im <= 2000000000.0:
		tmp = math.cos(re) + t_0
	elif im <= 3.1e+154:
		tmp = t_1
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(0.5 * Float64(im * im)))
	t_1 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 2000000000.0)
		tmp = Float64(cos(re) + t_0);
	elseif (im <= 3.1e+154)
		tmp = t_1;
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (0.5 * (im * im));
	t_1 = 0.5 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 2000000000.0)
		tmp = cos(re) + t_0;
	elseif (im <= 3.1e+154)
		tmp = t_1;
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * 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, -1.35e+154], t$95$0, If[LessEqual[im, -2.6], t$95$1, If[LessEqual[im, 2000000000.0], N[(N[Cos[re], $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[im, 3.1e+154], t$95$1, N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154

   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%

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

   if -1.35000000000000003e154 < im < -2.60000000000000009 or 2e9 < im < 3.1000000000000001e154

   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 72.6%

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

   if -2.60000000000000009 < im < 2e9

   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 98.0%

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

   4. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 98.0%

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

   6. Applied egg-rr 98.0%

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

   if 3.1000000000000001e154 < 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%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\cos re \cdot t_0\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im im))) (t_1 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im -1.35e+154)
     (* (cos re) t_0)
     (if (<= im -2.6)
       t_1
       (if (<= im 2000000000.0)
         (* (cos re) (+ t_0 1.0))
         (if (<= im 2.2e+154) t_1 (* im (* 0.5 (* (cos re) im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = cos(re) * t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 2000000000.0) {
		tmp = cos(re) * (t_0 + 1.0);
	} else if (im <= 2.2e+154) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (cos(re) * im));
	}
	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 = 0.5d0 * (im * im)
    t_1 = 0.5d0 * (exp(-im) + exp(im))
    if (im <= (-1.35d+154)) then
        tmp = cos(re) * t_0
    else if (im <= (-2.6d0)) then
        tmp = t_1
    else if (im <= 2000000000.0d0) then
        tmp = cos(re) * (t_0 + 1.0d0)
    else if (im <= 2.2d+154) then
        tmp = t_1
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * im);
	double t_1 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = Math.cos(re) * t_0;
	} else if (im <= -2.6) {
		tmp = t_1;
	} else if (im <= 2000000000.0) {
		tmp = Math.cos(re) * (t_0 + 1.0);
	} else if (im <= 2.2e+154) {
		tmp = t_1;
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * im)
	t_1 = 0.5 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= -1.35e+154:
		tmp = math.cos(re) * t_0
	elif im <= -2.6:
		tmp = t_1
	elif im <= 2000000000.0:
		tmp = math.cos(re) * (t_0 + 1.0)
	elif im <= 2.2e+154:
		tmp = t_1
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * im))
	t_1 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = Float64(cos(re) * t_0);
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 2000000000.0)
		tmp = Float64(cos(re) * Float64(t_0 + 1.0));
	elseif (im <= 2.2e+154)
		tmp = t_1;
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * im);
	t_1 = 0.5 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = cos(re) * t_0;
	elseif (im <= -2.6)
		tmp = t_1;
	elseif (im <= 2000000000.0)
		tmp = cos(re) * (t_0 + 1.0);
	elseif (im <= 2.2e+154)
		tmp = t_1;
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[im, -2.6], t$95$1, If[LessEqual[im, 2000000000.0], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.2e+154], t$95$1, N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154

   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%

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

   if -1.35000000000000003e154 < im < -2.60000000000000009 or 2e9 < im < 2.2000000000000001e154

   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 72.6%

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

   if -2.60000000000000009 < im < 2e9

   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 98.0%

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

   4. Simplified 98.0%

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

   5. Taylor expanded in re around inf 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 98.0%

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

      3. unpow2 98.0%

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

      4. distribute-lft1-in 98.0%

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

      5. +-commutative 98.0%

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

      6. unpow2 98.0%

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

      7. *-commutative 98.0%

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

      8. unpow2 98.0%

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

   7. Simplified 98.0%

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

   if 2.2000000000000001e154 < 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%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.6:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 78.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ t_1 := \left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im))))
        (t_1 (* (* re re) (* (* im im) -0.25))))
   (if (<= im -1.9e+154)
     t_0
     (if (<= im -1900000000000.0)
       t_1
       (if (<= im 1.1e+49) (cos re) (if (<= im 1.9e+150) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double t_1 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -1.9e+154) {
		tmp = t_0;
	} else if (im <= -1900000000000.0) {
		tmp = t_1;
	} else if (im <= 1.1e+49) {
		tmp = cos(re);
	} else if (im <= 1.9e+150) {
		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 = (re * re) * ((im * im) * (-0.25d0))
    if (im <= (-1.9d+154)) then
        tmp = t_0
    else if (im <= (-1900000000000.0d0)) then
        tmp = t_1
    else if (im <= 1.1d+49) then
        tmp = cos(re)
    else if (im <= 1.9d+150) 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 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -1.9e+154) {
		tmp = t_0;
	} else if (im <= -1900000000000.0) {
		tmp = t_1;
	} else if (im <= 1.1e+49) {
		tmp = Math.cos(re);
	} else if (im <= 1.9e+150) {
		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 = (re * re) * ((im * im) * -0.25)
	tmp = 0
	if im <= -1.9e+154:
		tmp = t_0
	elif im <= -1900000000000.0:
		tmp = t_1
	elif im <= 1.1e+49:
		tmp = math.cos(re)
	elif im <= 1.9e+150:
		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(Float64(re * re) * Float64(Float64(im * im) * -0.25))
	tmp = 0.0
	if (im <= -1.9e+154)
		tmp = t_0;
	elseif (im <= -1900000000000.0)
		tmp = t_1;
	elseif (im <= 1.1e+49)
		tmp = cos(re);
	elseif (im <= 1.9e+150)
		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 = (re * re) * ((im * im) * -0.25);
	tmp = 0.0;
	if (im <= -1.9e+154)
		tmp = t_0;
	elseif (im <= -1900000000000.0)
		tmp = t_1;
	elseif (im <= 1.1e+49)
		tmp = cos(re);
	elseif (im <= 1.9e+150)
		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[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.9e+154], t$95$0, If[LessEqual[im, -1900000000000.0], t$95$1, If[LessEqual[im, 1.1e+49], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.9e+150], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.8999999999999999e154 or 1.89999999999999995e150 < 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. unpow2 98.4%

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

      2. associate-*r* 98.4%

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

      3. associate-*r* 98.4%

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

      4. *-commutative 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   if -1.8999999999999999e154 < im < -1.9e12 or 1.1e49 < im < 1.89999999999999995e150

   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 7.3%

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

   4. Simplified 7.3%

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

   5. Taylor expanded in im around inf 7.3%

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

      1. unpow2 7.3%

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

      2. associate-*r* 6.0%

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

      3. associate-*r* 6.0%

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

      4. *-commutative 6.0%

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

      5. *-commutative 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. associate-*r* 27.4%

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

      2. fma-def 27.4%

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

      3. unpow2 27.4%

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

      4. associate-*r* 27.4%

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

   10. Simplified 27.4%

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

   11. Taylor expanded in re around inf 25.3%

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

       1. *-commutative 25.3%

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

       2. associate-*l* 25.3%

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

       3. unpow2 25.3%

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

       4. unpow2 25.3%

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

   13. Simplified 25.3%

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

   if -1.9e12 < im < 1.1e49

   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 89.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+150}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 78.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -6 \cdot 10^{+139}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) (* (* im im) -0.25))))
   (if (<= im -6e+139)
     (* (cos re) (* 0.5 (* im im)))
     (if (<= im -1900000000000.0)
       t_0
       (if (<= im 1e+51)
         (cos re)
         (if (<= im 2.1e+150) t_0 (* im (* 0.5 (* (cos re) im)))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -6e+139) {
		tmp = cos(re) * (0.5 * (im * im));
	} else if (im <= -1900000000000.0) {
		tmp = t_0;
	} else if (im <= 1e+51) {
		tmp = cos(re);
	} else if (im <= 2.1e+150) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 * (cos(re) * im));
	}
	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 = (re * re) * ((im * im) * (-0.25d0))
    if (im <= (-6d+139)) then
        tmp = cos(re) * (0.5d0 * (im * im))
    else if (im <= (-1900000000000.0d0)) then
        tmp = t_0
    else if (im <= 1d+51) then
        tmp = cos(re)
    else if (im <= 2.1d+150) then
        tmp = t_0
    else
        tmp = im * (0.5d0 * (cos(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -6e+139) {
		tmp = Math.cos(re) * (0.5 * (im * im));
	} else if (im <= -1900000000000.0) {
		tmp = t_0;
	} else if (im <= 1e+51) {
		tmp = Math.cos(re);
	} else if (im <= 2.1e+150) {
		tmp = t_0;
	} else {
		tmp = im * (0.5 * (Math.cos(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * ((im * im) * -0.25)
	tmp = 0
	if im <= -6e+139:
		tmp = math.cos(re) * (0.5 * (im * im))
	elif im <= -1900000000000.0:
		tmp = t_0
	elif im <= 1e+51:
		tmp = math.cos(re)
	elif im <= 2.1e+150:
		tmp = t_0
	else:
		tmp = im * (0.5 * (math.cos(re) * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * Float64(Float64(im * im) * -0.25))
	tmp = 0.0
	if (im <= -6e+139)
		tmp = Float64(cos(re) * Float64(0.5 * Float64(im * im)));
	elseif (im <= -1900000000000.0)
		tmp = t_0;
	elseif (im <= 1e+51)
		tmp = cos(re);
	elseif (im <= 2.1e+150)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * ((im * im) * -0.25);
	tmp = 0.0;
	if (im <= -6e+139)
		tmp = cos(re) * (0.5 * (im * im));
	elseif (im <= -1900000000000.0)
		tmp = t_0;
	elseif (im <= 1e+51)
		tmp = cos(re);
	elseif (im <= 2.1e+150)
		tmp = t_0;
	else
		tmp = im * (0.5 * (cos(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6e+139], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -1900000000000.0], t$95$0, If[LessEqual[im, 1e+51], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.1e+150], t$95$0, N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.9999999999999999e139

   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 88.0%

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

   4. Simplified 88.0%

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

   5. Taylor expanded in im around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*r* 88.0%

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

      3. unpow2 88.0%

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

   7. Simplified 88.0%

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

   if -5.9999999999999999e139 < im < -1.9e12 or 1e51 < im < 2.09999999999999998e150

   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 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in im around inf 5.5%

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

      1. unpow2 5.5%

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

      2. associate-*r* 5.5%

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

      3. associate-*r* 5.5%

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

      4. *-commutative 5.5%

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

      5. *-commutative 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in re around 0 27.5%

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

      1. associate-*r* 27.5%

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

      2. fma-def 27.5%

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

      3. unpow2 27.5%

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

      4. associate-*r* 27.5%

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

   10. Simplified 27.5%

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

   11. Taylor expanded in re around inf 25.8%

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

       1. *-commutative 25.8%

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

       2. associate-*l* 25.8%

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

       3. unpow2 25.8%

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

       4. unpow2 25.8%

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

   13. Simplified 25.8%

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

   if -1.9e12 < im < 1e51

   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 89.8%

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

   if 2.09999999999999998e150 < 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 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in im around inf 96.9%

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

      1. unpow2 96.9%

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

      2. associate-*r* 96.9%

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

      3. associate-*r* 96.9%

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

      4. *-commutative 96.9%

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

      5. *-commutative 96.9%

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

   7. Simplified 96.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6 \cdot 10^{+139}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -1900000000000:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 10^{+51}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+150}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 76.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $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 72.1%

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

4. Simplified 72.1%

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

5. Taylor expanded in re around inf 72.1%

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

   1. *-commutative 72.1%

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

   2. associate-*l* 72.1%

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

   3. unpow2 72.1%

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

   4. distribute-lft1-in 72.1%

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

   5. +-commutative 72.1%

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

   6. unpow2 72.1%

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

   7. *-commutative 72.1%

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

   8. unpow2 72.1%

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

7. Simplified 72.1%

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

8. Final simplification 72.1%

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

Alternative 8: 72.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ t_1 := \left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{if}\;im \leq -1.85 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2050000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))) (t_1 (* (* re re) (* (* im im) -0.25))))
   (if (<= im -1.85e+154)
     t_0
     (if (<= im -2050000000000.0)
       t_1
       (if (<= im 1.1e+49) (cos re) (if (<= im 1.65e+150) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -1.85e+154) {
		tmp = t_0;
	} else if (im <= -2050000000000.0) {
		tmp = t_1;
	} else if (im <= 1.1e+49) {
		tmp = cos(re);
	} else if (im <= 1.65e+150) {
		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 * im)
    t_1 = (re * re) * ((im * im) * (-0.25d0))
    if (im <= (-1.85d+154)) then
        tmp = t_0
    else if (im <= (-2050000000000.0d0)) then
        tmp = t_1
    else if (im <= 1.1d+49) then
        tmp = cos(re)
    else if (im <= 1.65d+150) 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 * im);
	double t_1 = (re * re) * ((im * im) * -0.25);
	double tmp;
	if (im <= -1.85e+154) {
		tmp = t_0;
	} else if (im <= -2050000000000.0) {
		tmp = t_1;
	} else if (im <= 1.1e+49) {
		tmp = Math.cos(re);
	} else if (im <= 1.65e+150) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	t_1 = (re * re) * ((im * im) * -0.25)
	tmp = 0
	if im <= -1.85e+154:
		tmp = t_0
	elif im <= -2050000000000.0:
		tmp = t_1
	elif im <= 1.1e+49:
		tmp = math.cos(re)
	elif im <= 1.65e+150:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	t_1 = Float64(Float64(re * re) * Float64(Float64(im * im) * -0.25))
	tmp = 0.0
	if (im <= -1.85e+154)
		tmp = t_0;
	elseif (im <= -2050000000000.0)
		tmp = t_1;
	elseif (im <= 1.1e+49)
		tmp = cos(re);
	elseif (im <= 1.65e+150)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	t_1 = (re * re) * ((im * im) * -0.25);
	tmp = 0.0;
	if (im <= -1.85e+154)
		tmp = t_0;
	elseif (im <= -2050000000000.0)
		tmp = t_1;
	elseif (im <= 1.1e+49)
		tmp = cos(re);
	elseif (im <= 1.65e+150)
		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 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.85e+154], t$95$0, If[LessEqual[im, -2050000000000.0], t$95$1, If[LessEqual[im, 1.1e+49], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.65e+150], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.84999999999999997e154 or 1.6499999999999999e150 < 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. unpow2 98.4%

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

      2. associate-*r* 98.4%

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

      3. associate-*r* 98.4%

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

      4. *-commutative 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in re around 0 76.1%

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

   if -1.84999999999999997e154 < im < -2.05e12 or 1.1e49 < im < 1.6499999999999999e150

   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 7.3%

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

   4. Simplified 7.3%

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

   5. Taylor expanded in im around inf 7.3%

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

      1. unpow2 7.3%

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

      2. associate-*r* 6.0%

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

      3. associate-*r* 6.0%

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

      4. *-commutative 6.0%

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

      5. *-commutative 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. associate-*r* 27.4%

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

      2. fma-def 27.4%

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

      3. unpow2 27.4%

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

      4. associate-*r* 27.4%

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

   10. Simplified 27.4%

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

   11. Taylor expanded in re around inf 25.3%

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

       1. *-commutative 25.3%

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

       2. associate-*l* 25.3%

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

       3. unpow2 25.3%

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

       4. unpow2 25.3%

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

   13. Simplified 25.3%

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

   if -2.05e12 < im < 1.1e49

   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 89.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.85 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq -2050000000000:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+49}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+150}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 9: 29.5% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ t_1 := im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))) (t_1 (* im (* 0.5 (* re re)))))
   (if (<= im -1.7e+154)
     t_0
     (if (<= im -2.5)
       t_1
       (if (<= im 1.5e+29) 0.25 (if (<= im 1.7e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = im * (0.5 * (re * re));
	double tmp;
	if (im <= -1.7e+154) {
		tmp = t_0;
	} else if (im <= -2.5) {
		tmp = t_1;
	} else if (im <= 1.5e+29) {
		tmp = 0.25;
	} else if (im <= 1.7e+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 * im)
    t_1 = im * (0.5d0 * (re * re))
    if (im <= (-1.7d+154)) then
        tmp = t_0
    else if (im <= (-2.5d0)) then
        tmp = t_1
    else if (im <= 1.5d+29) then
        tmp = 0.25d0
    else if (im <= 1.7d+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 * im);
	double t_1 = im * (0.5 * (re * re));
	double tmp;
	if (im <= -1.7e+154) {
		tmp = t_0;
	} else if (im <= -2.5) {
		tmp = t_1;
	} else if (im <= 1.5e+29) {
		tmp = 0.25;
	} else if (im <= 1.7e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	t_1 = im * (0.5 * (re * re))
	tmp = 0
	if im <= -1.7e+154:
		tmp = t_0
	elif im <= -2.5:
		tmp = t_1
	elif im <= 1.5e+29:
		tmp = 0.25
	elif im <= 1.7e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	t_1 = Float64(im * Float64(0.5 * Float64(re * re)))
	tmp = 0.0
	if (im <= -1.7e+154)
		tmp = t_0;
	elseif (im <= -2.5)
		tmp = t_1;
	elseif (im <= 1.5e+29)
		tmp = 0.25;
	elseif (im <= 1.7e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	t_1 = im * (0.5 * (re * re));
	tmp = 0.0;
	if (im <= -1.7e+154)
		tmp = t_0;
	elseif (im <= -2.5)
		tmp = t_1;
	elseif (im <= 1.5e+29)
		tmp = 0.25;
	elseif (im <= 1.7e+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 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.7e+154], t$95$0, If[LessEqual[im, -2.5], t$95$1, If[LessEqual[im, 1.5e+29], 0.25, If[LessEqual[im, 1.7e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.69999999999999987e154 or 1.69999999999999987e154 < 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%

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

   8. Taylor expanded in re around 0 77.4%

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

   if -1.69999999999999987e154 < im < -2.5 or 1.5e29 < im < 1.69999999999999987e154

   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 7.2%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in im around inf 7.2%

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

      1. unpow2 7.2%

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

      2. associate-*r* 6.0%

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

      3. associate-*r* 6.0%

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

      4. *-commutative 6.0%

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

      5. *-commutative 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in re around 0 24.5%

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

      1. associate-*r* 24.5%

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

      2. fma-def 24.5%

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

      3. unpow2 24.5%

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

      4. associate-*r* 24.5%

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

   10. Simplified 24.5%

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

   12. Applied egg-rr 18.7%

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

   if -2.5 < im < 1.5e29

   1. Initial program 100.0%

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

   3. Applied egg-rr 12.9%

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

   4. Taylor expanded in re around 0 13.0%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq -2.5:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 10: 48.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;im \cdot \left(-0.25 \cdot \left(re \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e+59)
   (+ (* 0.5 (* im im)) 1.0)
   (if (<= re 2.8e+195) (* im (* -0.25 (* re (* re im)))) (- (* re re) re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+59) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else if (re <= 2.8e+195) {
		tmp = im * (-0.25 * (re * (re * im)));
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.6d+59) then
        tmp = (0.5d0 * (im * im)) + 1.0d0
    else if (re <= 2.8d+195) then
        tmp = im * ((-0.25d0) * (re * (re * im)))
    else
        tmp = (re * re) - re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+59) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else if (re <= 2.8e+195) {
		tmp = im * (-0.25 * (re * (re * im)));
	} else {
		tmp = (re * re) - re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e+59:
		tmp = (0.5 * (im * im)) + 1.0
	elif re <= 2.8e+195:
		tmp = im * (-0.25 * (re * (re * im)))
	else:
		tmp = (re * re) - re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e+59)
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	elseif (re <= 2.8e+195)
		tmp = Float64(im * Float64(-0.25 * Float64(re * Float64(re * im))));
	else
		tmp = Float64(Float64(re * re) - re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.6e+59)
		tmp = (0.5 * (im * im)) + 1.0;
	elseif (re <= 2.8e+195)
		tmp = im * (-0.25 * (re * (re * im)));
	else
		tmp = (re * re) - re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e+59], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[re, 2.8e+195], N[(im * N[(-0.25 * N[(re * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.59999999999999991e59

   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 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in re around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

   7. Simplified 52.0%

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

   if 1.59999999999999991e59 < re < 2.7999999999999998e195

   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 63.7%

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

   4. Simplified 63.7%

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

   5. Taylor expanded in im around inf 26.1%

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

      1. unpow2 26.1%

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

      2. associate-*r* 26.1%

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

      3. associate-*r* 26.1%

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

      4. *-commutative 26.1%

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

      5. *-commutative 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in re around 0 38.7%

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

      1. associate-*r* 38.7%

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

      2. fma-def 38.7%

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

      3. unpow2 38.7%

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

      4. associate-*r* 38.7%

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

   10. Simplified 38.7%

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

   11. Taylor expanded in re around inf 38.6%

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

       1. *-commutative 38.6%

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

       2. unpow2 38.6%

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

       3. associate-*r* 38.7%

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

       4. *-commutative 38.7%

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

       5. associate-*l* 38.7%

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

       6. unpow2 38.7%

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

       7. associate-*l* 38.7%

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

       8. *-commutative 38.7%

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

   13. Simplified 38.7%

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

   if 2.7999999999999998e195 < 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 76.0%

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

   4. Simplified 76.0%

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

   5. Taylor expanded in im around inf 28.0%

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

      1. unpow2 28.0%

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

      2. associate-*r* 28.0%

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

      3. associate-*r* 28.0%

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

      4. *-commutative 28.0%

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

      5. *-commutative 28.0%

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

   7. Simplified 28.0%

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

   8. Taylor expanded in re around 0 0.9%

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

      1. associate-*r* 0.9%

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

      2. fma-def 0.9%

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

      3. unpow2 0.9%

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

      4. associate-*r* 0.9%

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

   10. Simplified 0.9%

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

   12. Applied egg-rr 50.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   13. Step-by-step derivation

       1. fma-neg 50.9%

          \[\leadsto \color{blue}{re \cdot re - re} \]

   14. Simplified 50.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{+195}:\\ \;\;\;\;im \cdot \left(-0.25 \cdot \left(re \cdot \left(re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot re - re\\ \end{array} \]

Alternative 11: 27.5% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.3 \cdot 10^{+28}:\\ \;\;\;\;re \cdot re - re\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))))
   (if (<= im -2.7e+151)
     t_0
     (if (<= im -2.3e+28)
       (- (* re re) re)
       (if (<= im 2000000000.0) 0.25 t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= -2.7e+151) {
		tmp = t_0;
	} else if (im <= -2.3e+28) {
		tmp = (re * re) - re;
	} else if (im <= 2000000000.0) {
		tmp = 0.25;
	} 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 * im)
    if (im <= (-2.7d+151)) then
        tmp = t_0
    else if (im <= (-2.3d+28)) then
        tmp = (re * re) - re
    else if (im <= 2000000000.0d0) then
        tmp = 0.25d0
    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 * im);
	double tmp;
	if (im <= -2.7e+151) {
		tmp = t_0;
	} else if (im <= -2.3e+28) {
		tmp = (re * re) - re;
	} else if (im <= 2000000000.0) {
		tmp = 0.25;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	tmp = 0
	if im <= -2.7e+151:
		tmp = t_0
	elif im <= -2.3e+28:
		tmp = (re * re) - re
	elif im <= 2000000000.0:
		tmp = 0.25
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= -2.7e+151)
		tmp = t_0;
	elseif (im <= -2.3e+28)
		tmp = Float64(Float64(re * re) - re);
	elseif (im <= 2000000000.0)
		tmp = 0.25;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	tmp = 0.0;
	if (im <= -2.7e+151)
		tmp = t_0;
	elseif (im <= -2.3e+28)
		tmp = (re * re) - re;
	elseif (im <= 2000000000.0)
		tmp = 0.25;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.7e+151], t$95$0, If[LessEqual[im, -2.3e+28], N[(N[(re * re), $MachinePrecision] - re), $MachinePrecision], If[LessEqual[im, 2000000000.0], 0.25, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.7000000000000001e151 or 2e9 < 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 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in im around inf 62.2%

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

      1. unpow2 62.2%

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

      2. associate-*r* 61.3%

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

      3. associate-*r* 61.3%

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

      4. *-commutative 61.3%

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

      5. *-commutative 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in re around 0 47.2%

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

   if -2.7000000000000001e151 < im < -2.29999999999999984e28

   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 5.9%

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

      1. unpow2 5.9%

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

      2. associate-*r* 5.9%

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

      3. associate-*r* 5.9%

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

      4. *-commutative 5.9%

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

      5. *-commutative 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in re around 0 22.6%

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

      1. associate-*r* 22.6%

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

      2. fma-def 22.6%

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

      3. unpow2 22.6%

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

      4. associate-*r* 22.6%

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

   10. Simplified 22.6%

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

   12. Applied egg-rr 17.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re, re, -re\right)} \]
   13. Step-by-step derivation

       1. fma-neg 17.4%

          \[\leadsto \color{blue}{re \cdot re - re} \]

   14. Simplified 17.4%

       \[\leadsto \color{blue}{re \cdot re - re} \]

   if -2.29999999999999984e28 < im < 2e9

   1. Initial program 100.0%

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

   3. Applied egg-rr 12.7%

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

   4. Taylor expanded in re around 0 12.8%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq -2.3 \cdot 10^{+28}:\\ \;\;\;\;re \cdot re - re\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 12: 27.2% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-0.5\right)\right)\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))))
   (if (<= im -1.45e+154)
     t_0
     (if (<= im -2.1e+14)
       (* im (* re (- 0.5)))
       (if (<= im 2000000000.0) 0.25 t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= -1.45e+154) {
		tmp = t_0;
	} else if (im <= -2.1e+14) {
		tmp = im * (re * -0.5);
	} else if (im <= 2000000000.0) {
		tmp = 0.25;
	} 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 * im)
    if (im <= (-1.45d+154)) then
        tmp = t_0
    else if (im <= (-2.1d+14)) then
        tmp = im * (re * -0.5d0)
    else if (im <= 2000000000.0d0) then
        tmp = 0.25d0
    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 * im);
	double tmp;
	if (im <= -1.45e+154) {
		tmp = t_0;
	} else if (im <= -2.1e+14) {
		tmp = im * (re * -0.5);
	} else if (im <= 2000000000.0) {
		tmp = 0.25;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	tmp = 0
	if im <= -1.45e+154:
		tmp = t_0
	elif im <= -2.1e+14:
		tmp = im * (re * -0.5)
	elif im <= 2000000000.0:
		tmp = 0.25
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= -1.45e+154)
		tmp = t_0;
	elseif (im <= -2.1e+14)
		tmp = Float64(im * Float64(re * Float64(-0.5)));
	elseif (im <= 2000000000.0)
		tmp = 0.25;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	tmp = 0.0;
	if (im <= -1.45e+154)
		tmp = t_0;
	elseif (im <= -2.1e+14)
		tmp = im * (re * -0.5);
	elseif (im <= 2000000000.0)
		tmp = 0.25;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.45e+154], t$95$0, If[LessEqual[im, -2.1e+14], N[(im * N[(re * (-0.5)), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2000000000.0], 0.25, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.4499999999999999e154 or 2e9 < 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 61.8%

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

   4. Simplified 61.8%

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

   5. Taylor expanded in im around inf 61.8%

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

      1. unpow2 61.8%

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

      2. associate-*r* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in re around 0 47.8%

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

   if -1.4499999999999999e154 < im < -2.1e14

   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 8.2%

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

   4. Simplified 8.2%

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

   5. Taylor expanded in im around inf 8.2%

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

      1. unpow2 8.2%

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

      2. associate-*r* 5.9%

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

      3. associate-*r* 5.9%

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

      4. *-commutative 5.9%

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

      5. *-commutative 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in re around 0 25.4%

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

      1. associate-*r* 25.4%

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

      2. fma-def 25.4%

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

      3. unpow2 25.4%

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

      4. associate-*r* 25.4%

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

   10. Simplified 25.4%

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

   12. Applied egg-rr 16.2%

       \[\leadsto im \cdot \left(0.5 \cdot \color{blue}{\left(-1 \cdot re\right)}\right) \]
   13. Step-by-step derivation

       1. neg-mul-1 16.2%

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

   14. Simplified 16.2%

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

   if -2.1e14 < im < 2e9

   1. Initial program 100.0%

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

   3. Applied egg-rr 13.0%

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

   4. Taylor expanded in re around 0 13.1%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq -2.1 \cdot 10^{+14}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-0.5\right)\right)\\ \mathbf{elif}\;im \leq 2000000000:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 13: 26.6% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.6 \lor \neg \left(im \leq 2000000000\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.6) (not (<= im 2000000000.0))) (* im (* 0.5 im)) 0.25))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.6) || !(im <= 2000000000.0)) {
		tmp = im * (0.5 * im);
	} 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 <= (-2.6d0)) .or. (.not. (im <= 2000000000.0d0))) then
        tmp = im * (0.5d0 * im)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.6) || !(im <= 2000000000.0)) {
		tmp = im * (0.5 * im);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.6) or not (im <= 2000000000.0):
		tmp = im * (0.5 * im)
	else:
		tmp = 0.25
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.6) || !(im <= 2000000000.0))
		tmp = Float64(im * Float64(0.5 * im));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.6) || ~((im <= 2000000000.0)))
		tmp = im * (0.5 * im);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.6], N[Not[LessEqual[im, 2000000000.0]], $MachinePrecision]], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision], 0.25]

Derivation

1. Split input into 2 regimes

2. if im < -2.60000000000000009 or 2e9 < 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 45.7%

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

   4. Simplified 45.7%

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

   5. Taylor expanded in im around inf 45.7%

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

      1. unpow2 45.7%

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

      2. associate-*r* 45.0%

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

      3. associate-*r* 45.0%

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

      4. *-commutative 45.0%

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

      5. *-commutative 45.0%

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

   7. Simplified 45.0%

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

   8. Taylor expanded in re around 0 34.6%

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

   if -2.60000000000000009 < im < 2e9

   1. Initial program 100.0%

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

   3. Applied egg-rr 13.2%

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

   4. Taylor expanded in re around 0 13.3%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \lor \neg \left(im \leq 2000000000\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 14: 48.2% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.6e+59) (+ (* 0.5 (* im im)) 1.0) (* im (* 0.5 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+59) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.6d+59) then
        tmp = (0.5d0 * (im * im)) + 1.0d0
    else
        tmp = im * (0.5d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.6e+59) {
		tmp = (0.5 * (im * im)) + 1.0;
	} else {
		tmp = im * (0.5 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.6e+59:
		tmp = (0.5 * (im * im)) + 1.0
	else:
		tmp = im * (0.5 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.6e+59)
		tmp = Float64(Float64(0.5 * Float64(im * im)) + 1.0);
	else
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.6e+59)
		tmp = (0.5 * (im * im)) + 1.0;
	else
		tmp = im * (0.5 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.6e+59], N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.59999999999999991e59

   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 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in re around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

   7. Simplified 52.0%

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

   if 1.59999999999999991e59 < 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 66.9%

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

   4. Simplified 66.9%

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

   5. Taylor expanded in im around inf 26.6%

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

      1. unpow2 26.6%

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

      2. associate-*r* 26.6%

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

      3. associate-*r* 26.6%

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

      4. *-commutative 26.6%

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

      5. *-commutative 26.6%

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

   7. Simplified 26.6%

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

   8. Taylor expanded in re around 0 28.6%

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

      1. associate-*r* 28.6%

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

      2. fma-def 28.6%

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

      3. unpow2 28.6%

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

      4. associate-*r* 28.6%

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

   10. Simplified 28.6%

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

   12. Applied egg-rr 26.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 15: 8.2% 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 7.9%

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

4. Taylor expanded in re around 0 7.9%

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

5. Final simplification 7.9%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))