math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 15

Speedup: 1.0×

• 49.6% 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: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1000:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1000.0)
   (* (cos re) (fma im (* 0.5 im) 1.0))
   (if (<= im 2.1e+154)
     (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))
     (* im (* (cos re) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1000.0) {
		tmp = cos(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 2.1e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im * (cos(re) * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1000.0)
		tmp = Float64(cos(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 2.1e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1000.0], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1e3

   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 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in re around inf 84.0%

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

      1. *-lft-identity 84.0%

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

      2. associate-*r* 84.0%

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

      3. distribute-rgt-out 84.0%

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

      4. +-commutative 84.0%

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

      5. *-commutative 84.0%

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

      6. unpow2 84.0%

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

      7. associate-*l* 84.0%

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

      8. fma-def 84.0%

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

   7. Simplified 84.0%

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

   if 1e3 < im < 2.09999999999999994e154

   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%

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

   4. Simplified 72.7%

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

   if 2.09999999999999994e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1000:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 71.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im 0.18)
     (cos re)
     (if (<= im 2.9e+100)
       t_0
       (if (<= im 5.4e+133)
         (* im (* im (+ 0.5 (* -0.25 (* re re)))))
         (if (<= im 1.9e+154) t_0 (* im (* (cos re) (* 0.5 im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= 0.18) {
		tmp = cos(re);
	} else if (im <= 2.9e+100) {
		tmp = t_0;
	} else if (im <= 5.4e+133) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else if (im <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = im * (cos(re) * (0.5 * 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 = 0.5d0 * (exp(-im) + exp(im))
    if (im <= 0.18d0) then
        tmp = cos(re)
    else if (im <= 2.9d+100) then
        tmp = t_0
    else if (im <= 5.4d+133) then
        tmp = im * (im * (0.5d0 + ((-0.25d0) * (re * re))))
    else if (im <= 1.9d+154) then
        tmp = t_0
    else
        tmp = im * (cos(re) * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (im <= 0.18) {
		tmp = Math.cos(re);
	} else if (im <= 2.9e+100) {
		tmp = t_0;
	} else if (im <= 5.4e+133) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else if (im <= 1.9e+154) {
		tmp = t_0;
	} else {
		tmp = im * (Math.cos(re) * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if im <= 0.18:
		tmp = math.cos(re)
	elif im <= 2.9e+100:
		tmp = t_0
	elif im <= 5.4e+133:
		tmp = im * (im * (0.5 + (-0.25 * (re * re))))
	elif im <= 1.9e+154:
		tmp = t_0
	else:
		tmp = im * (math.cos(re) * (0.5 * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= 0.18)
		tmp = cos(re);
	elseif (im <= 2.9e+100)
		tmp = t_0;
	elseif (im <= 5.4e+133)
		tmp = Float64(im * Float64(im * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	elseif (im <= 1.9e+154)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (im <= 0.18)
		tmp = cos(re);
	elseif (im <= 2.9e+100)
		tmp = t_0;
	elseif (im <= 5.4e+133)
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	elseif (im <= 1.9e+154)
		tmp = t_0;
	else
		tmp = im * (cos(re) * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.18], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.9e+100], t$95$0, If[LessEqual[im, 5.4e+133], N[(im * N[(im * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], t$95$0, N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.17999999999999999

   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 67.6%

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

   if 0.17999999999999999 < im < 2.9e100 or 5.4000000000000004e133 < im < 1.8999999999999999e154

   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 80.0%

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

   if 2.9e100 < im < 5.4000000000000004e133

   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 6.5%

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

   4. Simplified 6.5%

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

   5. Taylor expanded in im around inf 6.5%

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

      1. *-commutative 6.5%

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

      2. *-commutative 6.5%

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

      3. unpow2 6.5%

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

      4. associate-*r* 6.5%

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

      5. associate-*r* 6.5%

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

      6. *-commutative 6.5%

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

      7. associate-*l* 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*r* 79.2%

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

      3. distribute-rgt-out 79.2%

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

      4. unpow2 79.2%

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

   10. Simplified 79.2%

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

   if 1.8999999999999999e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ (exp (- im)) (exp im)))))
   (if (<= im 0.18)
     (* (cos re) (fma im (* 0.5 im) 1.0))
     (if (<= im 2.9e+100)
       t_0
       (if (<= im 5.4e+133)
         (* im (* im (+ 0.5 (* -0.25 (* re re)))))
         (if (<= im 2.5e+154) t_0 (* im (* (cos re) (* 0.5 im)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) + exp(im));
	double tmp;
	if (im <= 0.18) {
		tmp = cos(re) * fma(im, (0.5 * im), 1.0);
	} else if (im <= 2.9e+100) {
		tmp = t_0;
	} else if (im <= 5.4e+133) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else if (im <= 2.5e+154) {
		tmp = t_0;
	} else {
		tmp = im * (cos(re) * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (im <= 0.18)
		tmp = Float64(cos(re) * fma(im, Float64(0.5 * im), 1.0));
	elseif (im <= 2.9e+100)
		tmp = t_0;
	elseif (im <= 5.4e+133)
		tmp = Float64(im * Float64(im * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	elseif (im <= 2.5e+154)
		tmp = t_0;
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.18], N[(N[Cos[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.9e+100], t$95$0, If[LessEqual[im, 5.4e+133], N[(im * N[(im * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+154], t$95$0, N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 0.17999999999999999

   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 84.0%

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

   4. Simplified 84.0%

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

   5. Taylor expanded in re around inf 84.0%

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

      1. *-lft-identity 84.0%

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

      2. associate-*r* 84.0%

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

      3. distribute-rgt-out 84.0%

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

      4. +-commutative 84.0%

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

      5. *-commutative 84.0%

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

      6. unpow2 84.0%

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

      7. associate-*l* 84.0%

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

      8. fma-def 84.0%

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

   7. Simplified 84.0%

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

   if 0.17999999999999999 < im < 2.9e100 or 5.4000000000000004e133 < im < 2.50000000000000002e154

   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 80.0%

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

   if 2.9e100 < im < 5.4000000000000004e133

   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 6.5%

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

   4. Simplified 6.5%

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

   5. Taylor expanded in im around inf 6.5%

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

      1. *-commutative 6.5%

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

      2. *-commutative 6.5%

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

      3. unpow2 6.5%

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

      4. associate-*r* 6.5%

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

      5. associate-*r* 6.5%

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

      6. *-commutative 6.5%

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

      7. associate-*l* 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*r* 79.2%

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

      3. distribute-rgt-out 79.2%

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

      4. unpow2 79.2%

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

   10. Simplified 79.2%

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

   if 2.50000000000000002e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.18:\\ \;\;\;\;\cos re \cdot \mathsf{fma}\left(im, 0.5 \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+133}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 66.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot \left(im \cdot im\right)\\ t_1 := -0.25 \cdot t_0\\ t_2 := 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - t_2 \cdot t_2}{\mathsf{fma}\left(-0.25, t_0, 0.5 \cdot \left(im \cdot \left(-im\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) (* im im)))
        (t_1 (* -0.25 t_0))
        (t_2 (* 0.5 (* im im))))
   (if (<= im 12200000000000.0)
     (cos re)
     (if (<= im 4.5e+134)
       (+ (cos re) (* 0.5 (* (+ 1.0 (* (* re re) -0.5)) (* im im))))
       (if (<= im 1.4e+154)
         (/ (- (* t_1 t_1) (* t_2 t_2)) (fma -0.25 t_0 (* 0.5 (* im (- im)))))
         (* im (* (cos re) (* 0.5 im))))))))

C

double code(double re, double im) {
	double t_0 = (re * re) * (im * im);
	double t_1 = -0.25 * t_0;
	double t_2 = 0.5 * (im * im);
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = cos(re);
	} else if (im <= 4.5e+134) {
		tmp = cos(re) + (0.5 * ((1.0 + ((re * re) * -0.5)) * (im * im)));
	} else if (im <= 1.4e+154) {
		tmp = ((t_1 * t_1) - (t_2 * t_2)) / fma(-0.25, t_0, (0.5 * (im * -im)));
	} else {
		tmp = im * (cos(re) * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * Float64(im * im))
	t_1 = Float64(-0.25 * t_0)
	t_2 = Float64(0.5 * Float64(im * im))
	tmp = 0.0
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 4.5e+134)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(im * im))));
	elseif (im <= 1.4e+154)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(t_2 * t_2)) / fma(-0.25, t_0, Float64(0.5 * Float64(im * Float64(-im)))));
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 12200000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.5e+134], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-0.25 * t$95$0 + N[(0.5 * N[(im * (-im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.22e13

   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 66.6%

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

   if 1.22e13 < im < 4.4999999999999997e134

   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 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*r* 27.4%

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

      3. distribute-lft1-in 27.4%

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

      4. unpow2 27.4%

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

      5. unpow2 27.4%

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

   7. Simplified 27.4%

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

   if 4.4999999999999997e134 < im < 1.4e154

   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 10.5%

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

   4. Simplified 10.5%

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

   5. Taylor expanded in im around inf 10.5%

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

      1. *-commutative 10.5%

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

      2. *-commutative 10.5%

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

      3. unpow2 10.5%

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

      4. associate-*r* 10.5%

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

      5. associate-*r* 10.5%

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

      6. *-commutative 10.5%

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

      7. associate-*l* 10.5%

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

   7. Simplified 10.5%

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

   8. Taylor expanded in re around 0 8.4%

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

      1. *-commutative 8.4%

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

      2. associate-*r* 8.4%

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

      3. distribute-rgt-out 8.4%

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

      4. unpow2 8.4%

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

   10. Simplified 8.4%

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

       1. associate-*r* 8.4%

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

       2. distribute-lft-in 8.4%

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

       3. flip-+ 80.0%

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

       4. *-commutative 80.0%

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

       5. associate-*l* 80.0%

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

       6. *-commutative 80.0%

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

       7. associate-*l* 80.0%

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

       8. *-commutative 80.0%

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

       9. *-commutative 80.0%

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

       10. *-commutative 80.0%

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

       11. associate-*l* 80.0%

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

       12. *-commutative 80.0%

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

       13. fma-neg 80.0%

           \[\leadsto \frac{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{\color{blue}{\mathsf{fma}\left(-0.25, \left(re \cdot re\right) \cdot \left(im \cdot im\right), -0.5 \cdot \left(im \cdot im\right)\right)}} \]

   12. Applied egg-rr 80.0%

       \[\leadsto \color{blue}{\frac{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{\mathsf{fma}\left(-0.25, \left(re \cdot re\right) \cdot \left(im \cdot im\right), -0.5 \cdot \left(im \cdot im\right)\right)}} \]

   if 1.4e154 < 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%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(-0.25 \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{\mathsf{fma}\left(-0.25, \left(re \cdot re\right) \cdot \left(im \cdot im\right), 0.5 \cdot \left(im \cdot \left(-im\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 65.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+141}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12200000000000.0)
   (cos re)
   (if (<= im 3.5e+141)
     (+ (cos re) (* 0.5 (* (+ 1.0 (* (* re re) -0.5)) (* im im))))
     (* im (* (cos re) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = cos(re);
	} else if (im <= 3.5e+141) {
		tmp = cos(re) + (0.5 * ((1.0 + ((re * re) * -0.5)) * (im * im)));
	} else {
		tmp = im * (cos(re) * (0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 12200000000000.0d0) then
        tmp = cos(re)
    else if (im <= 3.5d+141) then
        tmp = cos(re) + (0.5d0 * ((1.0d0 + ((re * re) * (-0.5d0))) * (im * im)))
    else
        tmp = im * (cos(re) * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.5e+141) {
		tmp = Math.cos(re) + (0.5 * ((1.0 + ((re * re) * -0.5)) * (im * im)));
	} else {
		tmp = im * (Math.cos(re) * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12200000000000.0:
		tmp = math.cos(re)
	elif im <= 3.5e+141:
		tmp = math.cos(re) + (0.5 * ((1.0 + ((re * re) * -0.5)) * (im * im)))
	else:
		tmp = im * (math.cos(re) * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 3.5e+141)
		tmp = Float64(cos(re) + Float64(0.5 * Float64(Float64(1.0 + Float64(Float64(re * re) * -0.5)) * Float64(im * im))));
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 3.5e+141)
		tmp = cos(re) + (0.5 * ((1.0 + ((re * re) * -0.5)) * (im * im)));
	else
		tmp = im * (cos(re) * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12200000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.5e+141], N[(N[Cos[re], $MachinePrecision] + N[(0.5 * N[(N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.22e13

   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 66.6%

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

   if 1.22e13 < im < 3.5e141

   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 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in re around 0 26.8%

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

      1. *-commutative 26.8%

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

      2. associate-*r* 26.8%

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

      3. distribute-lft1-in 26.8%

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

      4. unpow2 26.8%

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

      5. unpow2 26.8%

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

   7. Simplified 26.8%

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

   if 3.5e141 < 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 87.3%

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

   4. Simplified 87.3%

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

   5. Taylor expanded in im around inf 87.3%

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

      1. *-commutative 87.3%

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

      2. *-commutative 87.3%

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

      3. unpow2 87.3%

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

      4. associate-*r* 87.3%

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

      5. associate-*r* 87.3%

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

      6. *-commutative 87.3%

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

      7. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+141}:\\ \;\;\;\;\cos re + 0.5 \cdot \left(\left(1 + \left(re \cdot re\right) \cdot -0.5\right) \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 65.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12200000000000.0)
   (cos re)
   (if (<= im 1.55e+135)
     (* im (* im (+ 0.5 (* -0.25 (* re re)))))
     (* im (* (cos re) (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = cos(re);
	} else if (im <= 1.55e+135) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else {
		tmp = im * (cos(re) * (0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 12200000000000.0d0) then
        tmp = cos(re)
    else if (im <= 1.55d+135) then
        tmp = im * (im * (0.5d0 + ((-0.25d0) * (re * re))))
    else
        tmp = im * (cos(re) * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.55e+135) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else {
		tmp = im * (Math.cos(re) * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12200000000000.0:
		tmp = math.cos(re)
	elif im <= 1.55e+135:
		tmp = im * (im * (0.5 + (-0.25 * (re * re))))
	else:
		tmp = im * (math.cos(re) * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 1.55e+135)
		tmp = Float64(im * Float64(im * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	else
		tmp = Float64(im * Float64(cos(re) * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 1.55e+135)
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	else
		tmp = im * (cos(re) * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12200000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.55e+135], N[(im * N[(im * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.22e13

   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 66.6%

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

   if 1.22e13 < im < 1.55000000000000011e135

   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 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around inf 4.8%

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

      1. *-commutative 4.8%

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

      2. *-commutative 4.8%

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

      3. unpow2 4.8%

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

      4. associate-*r* 4.8%

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

      5. associate-*r* 4.8%

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

      6. *-commutative 4.8%

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

      7. associate-*l* 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*r* 27.4%

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

      3. distribute-rgt-out 27.4%

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

      4. unpow2 27.4%

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

   10. Simplified 27.4%

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

   if 1.55000000000000011e135 < 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 84.6%

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

   4. Simplified 84.6%

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

   5. Taylor expanded in im around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. *-commutative 84.6%

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

      3. unpow2 84.6%

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

      4. associate-*r* 84.6%

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

      5. associate-*r* 84.6%

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

      6. *-commutative 84.6%

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

      7. associate-*l* 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 62.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12200000000000.0)
   (cos re)
   (if (<= im 1.55e+135)
     (* im (* im (+ 0.5 (* -0.25 (* re re)))))
     (+ 1.0 (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = cos(re);
	} else if (im <= 1.55e+135) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else {
		tmp = 1.0 + (im * (0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 12200000000000.0d0) then
        tmp = cos(re)
    else if (im <= 1.55d+135) then
        tmp = im * (im * (0.5d0 + ((-0.25d0) * (re * re))))
    else
        tmp = 1.0d0 + (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12200000000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.55e+135) {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	} else {
		tmp = 1.0 + (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12200000000000.0:
		tmp = math.cos(re)
	elif im <= 1.55e+135:
		tmp = im * (im * (0.5 + (-0.25 * (re * re))))
	else:
		tmp = 1.0 + (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 1.55e+135)
		tmp = Float64(im * Float64(im * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12200000000000.0)
		tmp = cos(re);
	elseif (im <= 1.55e+135)
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	else
		tmp = 1.0 + (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12200000000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.55e+135], N[(im * N[(im * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.22e13

   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 66.6%

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

   if 1.22e13 < im < 1.55000000000000011e135

   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 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around inf 4.8%

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

      1. *-commutative 4.8%

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

      2. *-commutative 4.8%

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

      3. unpow2 4.8%

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

      4. associate-*r* 4.8%

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

      5. associate-*r* 4.8%

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

      6. *-commutative 4.8%

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

      7. associate-*l* 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*r* 27.4%

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

      3. distribute-rgt-out 27.4%

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

      4. unpow2 27.4%

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

   10. Simplified 27.4%

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

   if 1.55000000000000011e135 < 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 84.6%

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

   4. Simplified 84.6%

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

   5. Taylor expanded in re around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. associate-*l* 67.3%

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

   7. Simplified 67.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+135}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 9: 49.1% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7} \lor \neg \left(im \leq 1.55 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 9.2e-7) (not (<= im 1.55e+135)))
   (+ 1.0 (* im (* 0.5 im)))
   (* im (* im (+ 0.5 (* -0.25 (* re re)))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 9.2e-7) || !(im <= 1.55e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = im * (im * (0.5 + (-0.25 * (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 ((im <= 9.2d-7) .or. (.not. (im <= 1.55d+135))) then
        tmp = 1.0d0 + (im * (0.5d0 * im))
    else
        tmp = im * (im * (0.5d0 + ((-0.25d0) * (re * re))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 9.2e-7) || !(im <= 1.55e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 9.2e-7) or not (im <= 1.55e+135):
		tmp = 1.0 + (im * (0.5 * im))
	else:
		tmp = im * (im * (0.5 + (-0.25 * (re * re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 9.2e-7) || !(im <= 1.55e+135))
		tmp = Float64(1.0 + Float64(im * Float64(0.5 * im)));
	else
		tmp = Float64(im * Float64(im * Float64(0.5 + Float64(-0.25 * Float64(re * re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 9.2e-7) || ~((im <= 1.55e+135)))
		tmp = 1.0 + (im * (0.5 * im));
	else
		tmp = im * (im * (0.5 + (-0.25 * (re * re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 9.2e-7], N[Not[LessEqual[im, 1.55e+135]], $MachinePrecision]], N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(im * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.1999999999999998e-7 or 1.55000000000000011e135 < 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 84.3%

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

   4. Simplified 84.3%

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

   5. Taylor expanded in re around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. unpow2 56.0%

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

      3. associate-*l* 56.0%

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

   7. Simplified 56.0%

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

   if 9.1999999999999998e-7 < im < 1.55000000000000011e135

   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 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in im around inf 4.9%

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

      1. *-commutative 4.9%

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

      2. *-commutative 4.9%

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

      3. unpow2 4.9%

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

      4. associate-*r* 4.9%

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

      5. associate-*r* 4.9%

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

      6. *-commutative 4.9%

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

      7. associate-*l* 4.9%

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

   7. Simplified 4.9%

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

   8. Taylor expanded in re around 0 24.9%

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

      1. *-commutative 24.9%

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

      2. associate-*r* 24.9%

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

      3. distribute-rgt-out 24.9%

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

      4. unpow2 24.9%

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

   10. Simplified 24.9%

       \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(-0.25 \cdot \left(re \cdot re\right) + 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7} \lor \neg \left(im \leq 1.55 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]

Alternative 10: 48.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12200000000000 \lor \neg \left(im \leq 1.55 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 12200000000000.0) (not (<= im 1.55e+135)))
   (+ 1.0 (* im (* 0.5 im)))
   (* -0.25 (* re (* re (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 12200000000000.0) || !(im <= 1.55e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = -0.25 * (re * (re * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 12200000000000.0d0) .or. (.not. (im <= 1.55d+135))) then
        tmp = 1.0d0 + (im * (0.5d0 * im))
    else
        tmp = (-0.25d0) * (re * (re * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 12200000000000.0) || !(im <= 1.55e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = -0.25 * (re * (re * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 12200000000000.0) or not (im <= 1.55e+135):
		tmp = 1.0 + (im * (0.5 * im))
	else:
		tmp = -0.25 * (re * (re * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 12200000000000.0) || !(im <= 1.55e+135))
		tmp = Float64(1.0 + Float64(im * Float64(0.5 * im)));
	else
		tmp = Float64(-0.25 * Float64(re * Float64(re * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 12200000000000.0) || ~((im <= 1.55e+135)))
		tmp = 1.0 + (im * (0.5 * im));
	else
		tmp = -0.25 * (re * (re * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 12200000000000.0], N[Not[LessEqual[im, 1.55e+135]], $MachinePrecision]], N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(re * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e13 or 1.55000000000000011e135 < 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 83.0%

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

   4. Simplified 83.0%

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

   5. Taylor expanded in re around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. unpow2 55.0%

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

      3. associate-*l* 55.0%

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

   7. Simplified 55.0%

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

   if 1.22e13 < im < 1.55000000000000011e135

   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 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around inf 4.8%

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

      1. *-commutative 4.8%

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

      2. *-commutative 4.8%

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

      3. unpow2 4.8%

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

      4. associate-*r* 4.8%

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

      5. associate-*r* 4.8%

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

      6. *-commutative 4.8%

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

      7. associate-*l* 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*r* 27.4%

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

      3. distribute-rgt-out 27.4%

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

      4. unpow2 27.4%

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

   10. Simplified 27.4%

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

   11. Taylor expanded in re around inf 26.1%

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

       1. unpow2 26.1%

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

       2. unpow2 26.1%

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

       3. *-commutative 26.1%

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

       4. associate-*l* 26.1%

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

   13. Simplified 26.1%

       \[\leadsto \color{blue}{-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000 \lor \neg \left(im \leq 1.55 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(re \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]

Alternative 11: 48.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12200000000000 \lor \neg \left(im \leq 1.05 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(im \cdot -0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 12200000000000.0) (not (<= im 1.05e+135)))
   (+ 1.0 (* im (* 0.5 im)))
   (* im (* re (* re (* im -0.25))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 12200000000000.0) || !(im <= 1.05e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = im * (re * (re * (im * -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 <= 12200000000000.0d0) .or. (.not. (im <= 1.05d+135))) then
        tmp = 1.0d0 + (im * (0.5d0 * im))
    else
        tmp = im * (re * (re * (im * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 12200000000000.0) || !(im <= 1.05e+135)) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = im * (re * (re * (im * -0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 12200000000000.0) or not (im <= 1.05e+135):
		tmp = 1.0 + (im * (0.5 * im))
	else:
		tmp = im * (re * (re * (im * -0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 12200000000000.0) || !(im <= 1.05e+135))
		tmp = Float64(1.0 + Float64(im * Float64(0.5 * im)));
	else
		tmp = Float64(im * Float64(re * Float64(re * Float64(im * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 12200000000000.0) || ~((im <= 1.05e+135)))
		tmp = 1.0 + (im * (0.5 * im));
	else
		tmp = im * (re * (re * (im * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 12200000000000.0], N[Not[LessEqual[im, 1.05e+135]], $MachinePrecision]], N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(im * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e13 or 1.05000000000000005e135 < 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 83.0%

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

   4. Simplified 83.0%

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

   5. Taylor expanded in re around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. unpow2 55.0%

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

      3. associate-*l* 55.0%

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

   7. Simplified 55.0%

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

   if 1.22e13 < im < 1.05000000000000005e135

   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 4.8%

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

   4. Simplified 4.8%

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

   5. Taylor expanded in im around inf 4.8%

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

      1. *-commutative 4.8%

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

      2. *-commutative 4.8%

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

      3. unpow2 4.8%

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

      4. associate-*r* 4.8%

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

      5. associate-*r* 4.8%

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

      6. *-commutative 4.8%

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

      7. associate-*l* 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in re around 0 27.4%

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

      1. *-commutative 27.4%

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

      2. associate-*r* 27.4%

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

      3. distribute-rgt-out 27.4%

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

      4. unpow2 27.4%

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

   10. Simplified 27.4%

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

   11. Taylor expanded in re around inf 26.1%

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

       1. unpow2 26.1%

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

       2. associate-*r* 26.1%

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

       3. *-commutative 26.1%

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

       4. associate-*l* 26.1%

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

       5. *-commutative 26.1%

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

   13. Simplified 26.1%

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(im \cdot -0.25\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12200000000000 \lor \neg \left(im \leq 1.05 \cdot 10^{+135}\right):\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(im \cdot -0.25\right)\right)\right)\\ \end{array} \]

Alternative 12: 19.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 310:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 310.0)
   0.25
   (if (<= im 4.5e+149) (+ 0.25 (* (* re re) 0.25)) (* im (* 0.5 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 310.0) {
		tmp = 0.25;
	} else if (im <= 4.5e+149) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 310.0d0) then
        tmp = 0.25d0
    else if (im <= 4.5d+149) then
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 310.0) {
		tmp = 0.25;
	} else if (im <= 4.5e+149) {
		tmp = 0.25 + ((re * re) * 0.25);
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 310.0:
		tmp = 0.25
	elif im <= 4.5e+149:
		tmp = 0.25 + ((re * re) * 0.25)
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 310.0)
		tmp = 0.25;
	elseif (im <= 4.5e+149)
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 310.0)
		tmp = 0.25;
	elseif (im <= 4.5e+149)
		tmp = 0.25 + ((re * re) * 0.25);
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 310.0], 0.25, If[LessEqual[im, 4.5e+149], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 310

   1. Initial program 100.0%

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

   3. Applied egg-rr 9.9%

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

   4. Taylor expanded in re around 0 10.0%

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

   if 310 < im < 4.49999999999999982e149

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.3%

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

   4. Taylor expanded in re around 0 14.0%

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

      1. *-commutative 14.0%

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

      2. unpow2 14.0%

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

   6. Simplified 14.0%

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

   if 4.49999999999999982e149 < 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 93.2%

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

   4. Simplified 93.2%

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

   5. Taylor expanded in im around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. *-commutative 93.2%

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

      3. unpow2 93.2%

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

      4. associate-*r* 93.2%

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

      5. associate-*r* 93.2%

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

      6. *-commutative 93.2%

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

      7. associate-*l* 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in re around 0 74.0%

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

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 310:\\ \;\;\;\;0.25\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 13: 47.9% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+166}:\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7e+166) (+ 1.0 (* im (* 0.5 im))) (+ 0.25 (* (* re re) 0.25))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7e+166) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = 0.25 + ((re * re) * 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 (re <= 7d+166) then
        tmp = 1.0d0 + (im * (0.5d0 * im))
    else
        tmp = 0.25d0 + ((re * re) * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7e+166) {
		tmp = 1.0 + (im * (0.5 * im));
	} else {
		tmp = 0.25 + ((re * re) * 0.25);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7e+166:
		tmp = 1.0 + (im * (0.5 * im))
	else:
		tmp = 0.25 + ((re * re) * 0.25)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7e+166)
		tmp = Float64(1.0 + Float64(im * Float64(0.5 * im)));
	else
		tmp = Float64(0.25 + Float64(Float64(re * re) * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7e+166)
		tmp = 1.0 + (im * (0.5 * im));
	else
		tmp = 0.25 + ((re * re) * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7e+166], N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(N[(re * re), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.9999999999999997e166

   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 71.8%

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

   4. Simplified 71.8%

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

   5. Taylor expanded in re around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. associate-*l* 51.5%

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

   7. Simplified 51.5%

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

   if 6.9999999999999997e166 < re

   1. Initial program 100.0%

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

   3. Applied egg-rr 6.0%

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

   4. Taylor expanded in re around 0 18.9%

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

      1. *-commutative 18.9%

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

      2. unpow2 18.9%

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

   6. Simplified 18.9%

      \[\leadsto \color{blue}{0.25 + \left(re \cdot re\right) \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+166}:\\ \;\;\;\;1 + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \]

Alternative 14: 17.8% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 9.2e-7) 0.25 (* im (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = 0.25;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 9.2d-7) then
        tmp = 0.25d0
    else
        tmp = im * (0.5d0 * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.2e-7) {
		tmp = 0.25;
	} else {
		tmp = im * (0.5 * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e-7:
		tmp = 0.25
	else:
		tmp = im * (0.5 * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e-7)
		tmp = 0.25;
	else
		tmp = Float64(im * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.2e-7)
		tmp = 0.25;
	else
		tmp = im * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e-7], 0.25, N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.1999999999999998e-7

   1. Initial program 100.0%

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

   3. Applied egg-rr 10.0%

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

   4. Taylor expanded in re around 0 10.0%

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

   if 9.1999999999999998e-7 < 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 38.8%

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

   4. Simplified 38.8%

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

   5. Taylor expanded in im around inf 38.4%

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

      1. *-commutative 38.4%

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

      2. *-commutative 38.4%

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

      3. unpow2 38.4%

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

      4. associate-*r* 38.4%

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

      5. associate-*r* 38.4%

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

      6. *-commutative 38.4%

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

      7. associate-*l* 38.4%

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

   7. Simplified 38.4%

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

   8. Taylor expanded in re around 0 30.0%

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

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;0.25\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 15: 8.1% 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 8.0%

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

5. Final simplification 8.0%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))