math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 9

Speedup: 1.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 11.2 \lor \neg \left(im \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 11.2) (not (<= im 1.15e+77)))
   (*
    (* 0.5 (cos re))
    (+ (+ 2.0 (* im im)) (* 0.08333333333333333 (pow im 4.0))))
   (* 0.5 (* (exp im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 11.2) || !(im <= 1.15e+77)) {
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * pow(im, 4.0)));
	} else {
		tmp = 0.5 * (exp(im) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 11.2d0) .or. (.not. (im <= 1.15d+77))) then
        tmp = (0.5d0 * cos(re)) * ((2.0d0 + (im * im)) + (0.08333333333333333d0 * (im ** 4.0d0)))
    else
        tmp = 0.5d0 * (exp(im) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 11.2) || !(im <= 1.15e+77)) {
		tmp = (0.5 * Math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * Math.pow(im, 4.0)));
	} else {
		tmp = 0.5 * (Math.exp(im) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 11.2) or not (im <= 1.15e+77):
		tmp = (0.5 * math.cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * math.pow(im, 4.0)))
	else:
		tmp = 0.5 * (math.exp(im) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 11.2) || !(im <= 1.15e+77))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64(0.08333333333333333 * (im ^ 4.0))));
	else
		tmp = Float64(0.5 * Float64(exp(im) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 11.2) || ~((im <= 1.15e+77)))
		tmp = (0.5 * cos(re)) * ((2.0 + (im * im)) + (0.08333333333333333 * (im ^ 4.0)));
	else
		tmp = 0.5 * (exp(im) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 11.2], N[Not[LessEqual[im, 1.15e+77]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(0.08333333333333333 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 11.199999999999999 or 1.14999999999999997e77 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 92.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 92.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

      2. unpow2 92.1%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + \color{blue}{im \cdot im}\right) + 0.08333333333333333 \cdot {im}^{4}\right) \]

   4. Simplified 92.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]

   if 11.199999999999999 < im < 1.14999999999999997e77

   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 88.9%

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

      1. *-commutative 88.9%

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

   4. Simplified 88.9%

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

      1. *-un-lft-identity 88.9%

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

      2. fma-def 88.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{-im}\right)} \cdot 0.5 \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}\right) \cdot 0.5 \]

      4. sqrt-unprod 88.9%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}}\right) \cdot 0.5 \]

      5. sqr-neg 88.9%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\sqrt{\color{blue}{im \cdot im}}}\right) \cdot 0.5 \]

      6. sqrt-prod 88.9%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) \cdot 0.5 \]

      7. add-sqr-sqrt 88.9%

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

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{im}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. fma-udef 88.9%

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

      2. distribute-lft1-in 88.9%

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

      3. metadata-eval 88.9%

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

   8. Simplified 88.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 11.2 \lor \neg \left(im \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(2 + im \cdot im\right) + 0.08333333333333333 \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \]

Alternative 3: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 6:\\ \;\;\;\;\cos re + t_0 \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 6.0)
     (+ (cos re) (* t_0 (* im im)))
     (if (<= im 1.35e+154)
       (* 0.5 (* (exp im) 2.0))
       (* t_0 (+ 2.0 (* im im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 6.0) {
		tmp = cos(re) + (t_0 * (im * im));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(im) * 2.0);
	} else {
		tmp = t_0 * (2.0 + (im * 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 * cos(re)
    if (im <= 6.0d0) then
        tmp = cos(re) + (t_0 * (im * im))
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(im) * 2.0d0)
    else
        tmp = t_0 * (2.0d0 + (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 6.0) {
		tmp = Math.cos(re) + (t_0 * (im * im));
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(im) * 2.0);
	} else {
		tmp = t_0 * (2.0 + (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 6.0:
		tmp = math.cos(re) + (t_0 * (im * im))
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(im) * 2.0)
	else:
		tmp = t_0 * (2.0 + (im * im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 6.0)
		tmp = Float64(cos(re) + Float64(t_0 * Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(im) * 2.0));
	else
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 6.0)
		tmp = cos(re) + (t_0 * (im * im));
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(im) * 2.0);
	else
		tmp = t_0 * (2.0 + (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 6.0], N[(N[Cos[re], $MachinePrecision] + N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 6

   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 81.1%

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

      1. associate-*r* 81.1%

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

      2. *-commutative 81.1%

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

      3. fma-def 81.1%

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

      4. unpow2 81.1%

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

   4. Simplified 81.1%

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

      1. fma-udef 81.1%

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

      2. *-commutative 81.1%

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

   6. Applied egg-rr 81.1%

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

   if 6 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 79.5%

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

      1. *-commutative 79.5%

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

   4. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. fma-def 79.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{-im}\right)} \cdot 0.5 \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}\right) \cdot 0.5 \]

      4. sqrt-unprod 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}}\right) \cdot 0.5 \]

      5. sqr-neg 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\sqrt{\color{blue}{im \cdot im}}}\right) \cdot 0.5 \]

      6. sqrt-prod 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) \cdot 0.5 \]

      7. add-sqr-sqrt 79.5%

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

   6. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{im}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. fma-udef 79.5%

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

      2. distribute-lft1-in 79.5%

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

      3. metadata-eval 79.5%

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

   8. Simplified 79.5%

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

   if 1.35000000000000003e154 < 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%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6:\\ \;\;\;\;\cos re + \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 4: 85.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.2 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 7.2) (not (<= im 1.35e+154)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (* 0.5 (* (exp im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 7.2) || !(im <= 1.35e+154)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (exp(im) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 7.2d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * (exp(im) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 7.2) || !(im <= 1.35e+154)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (Math.exp(im) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 7.2) or not (im <= 1.35e+154):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = 0.5 * (math.exp(im) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 7.2) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(exp(im) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 7.2) || ~((im <= 1.35e+154)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = 0.5 * (exp(im) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 7.2], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.20000000000000018 or 1.35000000000000003e154 < 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 84.7%

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

      1. unpow2 84.7%

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

   4. Simplified 84.7%

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

   if 7.20000000000000018 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 79.5%

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

      1. *-commutative 79.5%

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

   4. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. fma-def 79.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{-im}\right)} \cdot 0.5 \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}\right) \cdot 0.5 \]

      4. sqrt-unprod 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}}\right) \cdot 0.5 \]

      5. sqr-neg 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\sqrt{\color{blue}{im \cdot im}}}\right) \cdot 0.5 \]

      6. sqrt-prod 79.5%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) \cdot 0.5 \]

      7. add-sqr-sqrt 79.5%

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

   6. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{im}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. fma-udef 79.5%

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

      2. distribute-lft1-in 79.5%

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

      3. metadata-eval 79.5%

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

   8. Simplified 79.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.2 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \]

Alternative 5: 78.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7) (+ (cos re) (* im (* 0.5 im))) (* 0.5 (* (exp im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = cos(re) + (im * (0.5 * im));
	} else {
		tmp = 0.5 * (exp(im) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.7d0) then
        tmp = cos(re) + (im * (0.5d0 * im))
    else
        tmp = 0.5d0 * (exp(im) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7) {
		tmp = Math.cos(re) + (im * (0.5 * im));
	} else {
		tmp = 0.5 * (Math.exp(im) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7:
		tmp = math.cos(re) + (im * (0.5 * im))
	else:
		tmp = 0.5 * (math.exp(im) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7)
		tmp = Float64(cos(re) + Float64(im * Float64(0.5 * im)));
	else
		tmp = Float64(0.5 * Float64(exp(im) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7)
		tmp = cos(re) + (im * (0.5 * im));
	else
		tmp = 0.5 * (exp(im) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7], N[(N[Cos[re], $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7000000000000002

   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 81.1%

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

      1. associate-*r* 81.1%

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

      2. *-commutative 81.1%

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

      3. fma-def 81.1%

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

      4. unpow2 81.1%

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

   4. Simplified 81.1%

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

      1. fma-udef 81.1%

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

      2. *-commutative 81.1%

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

   6. Applied egg-rr 81.1%

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

   7. Taylor expanded in re around 0 77.9%

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

      1. unpow2 77.9%

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

      2. associate-*r* 77.9%

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

   9. Simplified 77.9%

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

   if 3.7000000000000002 < im

   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 76.3%

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

      1. *-commutative 76.3%

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

   4. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. fma-def 76.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{-im}\right)} \cdot 0.5 \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}\right) \cdot 0.5 \]

      4. sqrt-unprod 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}}\right) \cdot 0.5 \]

      5. sqr-neg 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\sqrt{\color{blue}{im \cdot im}}}\right) \cdot 0.5 \]

      6. sqrt-prod 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) \cdot 0.5 \]

      7. add-sqr-sqrt 76.3%

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

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{im}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. fma-udef 76.3%

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

      2. distribute-lft1-in 76.3%

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

      3. metadata-eval 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \]

Alternative 6: 68.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6) (cos re) (* 0.5 (* (exp im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(im) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.6d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(im) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(im) * 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(im) * 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(im) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(im) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.60000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 62.2%

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

   if 3.60000000000000009 < im

   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 76.3%

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

      1. *-commutative 76.3%

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

   4. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. fma-def 76.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{-im}\right)} \cdot 0.5 \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}\right) \cdot 0.5 \]

      4. sqrt-unprod 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}}\right) \cdot 0.5 \]

      5. sqr-neg 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\sqrt{\color{blue}{im \cdot im}}}\right) \cdot 0.5 \]

      6. sqrt-prod 76.3%

         \[\leadsto \mathsf{fma}\left(1, e^{im}, e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}\right) \cdot 0.5 \]

      7. add-sqr-sqrt 76.3%

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

   6. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, e^{im}, e^{im}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. fma-udef 76.3%

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

      2. distribute-lft1-in 76.3%

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

      3. metadata-eval 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} \cdot 2\right)\\ \end{array} \]

Alternative 7: 60.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right) + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.1e+39) (cos re) (+ (* im (* 0.5 im)) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.1e+39) {
		tmp = cos(re);
	} else {
		tmp = (im * (0.5 * im)) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.1d+39) then
        tmp = cos(re)
    else
        tmp = (im * (0.5d0 * im)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.1e+39) {
		tmp = Math.cos(re);
	} else {
		tmp = (im * (0.5 * im)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.1e+39:
		tmp = math.cos(re)
	else:
		tmp = (im * (0.5 * im)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.1e+39)
		tmp = cos(re);
	else
		tmp = Float64(Float64(im * Float64(0.5 * im)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.1e+39)
		tmp = cos(re);
	else
		tmp = (im * (0.5 * im)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.1e+39], N[Cos[re], $MachinePrecision], N[(N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.10000000000000004e39

   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 58.7%

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

   if 4.10000000000000004e39 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 61.9%

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

      1. associate-*r* 61.9%

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

      2. *-commutative 61.9%

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

      3. fma-def 61.9%

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

      4. unpow2 61.9%

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

   4. Simplified 61.9%

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

      1. fma-udef 61.9%

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

      2. *-commutative 61.9%

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

   6. Applied egg-rr 61.9%

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

   7. Taylor expanded in re around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 45.3%

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

   9. Simplified 45.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right) + 1\\ \end{array} \]

Alternative 8: 47.3% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. *-commutative 72.6%

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

   3. fma-def 72.6%

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

   4. unpow2 72.6%

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

4. Simplified 72.6%

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

   1. fma-udef 72.6%

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

   2. *-commutative 72.6%

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

6. Applied egg-rr 72.6%

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

7. Taylor expanded in re around 0 46.1%

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

   1. +-commutative 46.1%

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

   2. unpow2 46.1%

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

   3. associate-*r* 46.1%

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

9. Simplified 46.1%

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

10. Final simplification 46.1%

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

Alternative 9: 28.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 23.7%

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

   1. +-inverses 23.7%

      \[\leadsto \frac{-2 \cdot \cos re}{-2 \cdot \cos re + \color{blue}{0}} \]

   2. +-rgt-identity 23.7%

      \[\leadsto \frac{-2 \cdot \cos re}{\color{blue}{-2 \cdot \cos re}} \]

   3. *-inverses 23.7%

      \[\leadsto \color{blue}{1} \]

5. Simplified 23.7%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 23.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))