math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 10

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00062)
   (fma (* 0.5 im) im (cos re))
   (if (<= im 1.4e+154)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00062) {
		tmp = fma((0.5 * im), im, cos(re));
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00062)
		tmp = fma(Float64(0.5 * im), im, cos(re));
	elseif (im <= 1.4e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00062], N[(N[(0.5 * im), $MachinePrecision] * im + N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.2e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

   4. Taylor expanded in re around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r* 79.4%

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

      4. fma-def 79.4%

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

   6. Applied egg-rr 79.4%

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

   if 6.2e-4 < im < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 91.4%

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

   if 1.4e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\cos re \cdot \left(t\_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 0.00062)
     (* (cos re) (+ t_0 1.0))
     (if (<= im 1.4e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* (cos re) t_0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(im, 2.0);
	double tmp;
	if (im <= 0.00062) {
		tmp = cos(re) * (t_0 + 1.0);
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im ** 2.0d0)
    if (im <= 0.00062d0) then
        tmp = cos(re) * (t_0 + 1.0d0)
    else if (im <= 1.4d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(im, 2.0);
	double tmp;
	if (im <= 0.00062) {
		tmp = Math.cos(re) * (t_0 + 1.0);
	} else if (im <= 1.4e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(im, 2.0)
	tmp = 0
	if im <= 0.00062:
		tmp = math.cos(re) * (t_0 + 1.0)
	elif im <= 1.4e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (im ^ 2.0))
	tmp = 0.0
	if (im <= 0.00062)
		tmp = Float64(cos(re) * Float64(t_0 + 1.0));
	elseif (im <= 1.4e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im ^ 2.0);
	tmp = 0.0;
	if (im <= 0.00062)
		tmp = cos(re) * (t_0 + 1.0);
	elseif (im <= 1.4e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.00062], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.4e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 6.2e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

   5. Simplified 85.1%

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

   if 6.2e-4 < im < 1.4e154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 91.4%

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

   if 1.4e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9000 \lor \neg \left(im \leq 3.05 \cdot 10^{+128}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 9000.0) (not (<= im 3.05e+128)))
   (fma (* 0.5 im) im (cos re))
   (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 9000.0) || !(im <= 3.05e+128)) {
		tmp = fma((0.5 * im), im, cos(re));
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 9000.0) || !(im <= 3.05e+128))
		tmp = fma(Float64(0.5 * im), im, cos(re));
	else
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 9000.0], N[Not[LessEqual[im, 3.05e+128]], $MachinePrecision]], N[(N[(0.5 * im), $MachinePrecision] * im + N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9e3 or 3.0500000000000001e128 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.8%

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

   4. Taylor expanded in re around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. unpow2 75.3%

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

      3. associate-*r* 75.3%

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

      4. fma-def 75.3%

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

   6. Applied egg-rr 75.3%

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

   if 9e3 < im < 3.0500000000000001e128

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.8%

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

   5. Taylor expanded in re around 0 17.8%

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

      1. *-commutative 17.8%

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

   7. Simplified 17.8%

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

      1. unpow2 17.8%

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

   9. Applied egg-rr 17.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9000 \lor \neg \left(im \leq 3.05 \cdot 10^{+128}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00062)
   (fma (* 0.5 im) im (cos re))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00062) {
		tmp = fma((0.5 * im), im, cos(re));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00062)
		tmp = fma(Float64(0.5 * im), im, cos(re));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00062], N[(N[(0.5 * im), $MachinePrecision] * im + N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.2e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

   4. Taylor expanded in re around 0 79.4%

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

      1. +-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r* 79.4%

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

      4. fma-def 79.4%

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

   6. Applied egg-rr 79.4%

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

   if 6.2e-4 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 78.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00062:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+128}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7000.0)
   (cos re)
   (if (<= im 3.4e+128)
     (+ 0.25 (* 0.25 (* re re)))
     (+ (* 0.5 (pow im 2.0)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7000.0) {
		tmp = cos(re);
	} else if (im <= 3.4e+128) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = (0.5 * pow(im, 2.0)) + 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 <= 7000.0d0) then
        tmp = cos(re)
    else if (im <= 3.4d+128) then
        tmp = 0.25d0 + (0.25d0 * (re * re))
    else
        tmp = (0.5d0 * (im ** 2.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7000.0) {
		tmp = Math.cos(re);
	} else if (im <= 3.4e+128) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = (0.5 * Math.pow(im, 2.0)) + 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7000.0:
		tmp = math.cos(re)
	elif im <= 3.4e+128:
		tmp = 0.25 + (0.25 * (re * re))
	else:
		tmp = (0.5 * math.pow(im, 2.0)) + 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7000.0)
		tmp = cos(re);
	elseif (im <= 3.4e+128)
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	else
		tmp = Float64(Float64(0.5 * (im ^ 2.0)) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7000.0)
		tmp = cos(re);
	elseif (im <= 3.4e+128)
		tmp = 0.25 + (0.25 * (re * re));
	else
		tmp = (0.5 * (im ^ 2.0)) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 3.4e+128], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 7e3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.2%

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

   if 7e3 < im < 3.3999999999999999e128

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.8%

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

   5. Taylor expanded in re around 0 17.8%

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

      1. *-commutative 17.8%

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

   7. Simplified 17.8%

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

      1. unpow2 17.8%

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

   9. Applied egg-rr 17.8%

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

   if 3.3999999999999999e128 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

   5. Simplified 85.1%

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

   6. Taylor expanded in re around 0 55.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 3.4 \cdot 10^{+128}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 440000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+128}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 440000.0)
   (cos re)
   (if (<= im 2.7e+128) (+ 0.25 (* 0.25 (* re re))) (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 440000.0) {
		tmp = cos(re);
	} else if (im <= 2.7e+128) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = 0.5 * pow(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 <= 440000.0d0) then
        tmp = cos(re)
    else if (im <= 2.7d+128) then
        tmp = 0.25d0 + (0.25d0 * (re * re))
    else
        tmp = 0.5d0 * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 440000.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.7e+128) {
		tmp = 0.25 + (0.25 * (re * re));
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 440000.0:
		tmp = math.cos(re)
	elif im <= 2.7e+128:
		tmp = 0.25 + (0.25 * (re * re))
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 440000.0)
		tmp = cos(re);
	elseif (im <= 2.7e+128)
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 440000.0)
		tmp = cos(re);
	elseif (im <= 2.7e+128)
		tmp = 0.25 + (0.25 * (re * re));
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 440000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.7e+128], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.4e5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.2%

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

   if 4.4e5 < im < 2.70000000000000001e128

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.8%

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

   5. Taylor expanded in re around 0 17.8%

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

      1. *-commutative 17.8%

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

   7. Simplified 17.8%

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

      1. unpow2 17.8%

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

   9. Applied egg-rr 17.8%

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

   if 2.70000000000000001e128 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 85.1%

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

   4. Taylor expanded in re around 0 55.4%

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

   5. Taylor expanded in im around inf 55.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+128}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 12000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 12000.0) (cos re) (+ 0.25 (* 0.25 (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 12000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.25 + (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 <= 12000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.25d0 + (0.25d0 * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 12000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 + (0.25 * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 12000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.25 + (0.25 * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 12000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.25 + Float64(0.25 * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 12000.0)
		tmp = cos(re);
	else
		tmp = 0.25 + (0.25 * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 12000.0], N[Cos[re], $MachinePrecision], N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 12000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.2%

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

   if 12000 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 2.6%

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

   5. Taylor expanded in re around 0 13.3%

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

      1. *-commutative 13.3%

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

   7. Simplified 13.3%

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

      1. unpow2 13.3%

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

   9. Applied egg-rr 13.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 12000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 + 0.25 \cdot \left(re \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 12.4% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 0.25 + 0.25 \cdot \left(re \cdot re\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 0.25 (* 0.25 (* re re))))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25 + (0.25 * (re * re))

Julia

function code(re, im)
	return Float64(0.25 + Float64(0.25 * Float64(re * re)))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(0.25 + N[(0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in re around 0 13.4%

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

   1. *-commutative 13.4%

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

7. Simplified 13.4%

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

   1. unpow2 13.4%

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

9. Applied egg-rr 13.4%

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

10. Final simplification 13.4%

    \[\leadsto 0.25 + 0.25 \cdot \left(re \cdot re\right) \]
11. Add Preprocessing

Alternative 10: 8.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 7.9%

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

5. Taylor expanded in re around 0 8.0%

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

6. Final simplification 8.0%

   \[\leadsto 0.25 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))