math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 11

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. Final simplification 100.0%

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

Alternative 2: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.45e-28)
   (cos re)
   (if (<= im 2e+152)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45e-28) {
		tmp = cos(re);
	} else if (im <= 2e+152) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = (0.5 * cos(re)) * 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 <= 1.45d-28) then
        tmp = cos(re)
    else if (im <= 2d+152) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = (0.5d0 * cos(re)) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45e-28) {
		tmp = Math.cos(re);
	} else if (im <= 2e+152) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45e-28:
		tmp = math.cos(re)
	elif im <= 2e+152:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45e-28)
		tmp = cos(re);
	elseif (im <= 2e+152)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45e-28)
		tmp = cos(re);
	elseif (im <= 2e+152)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = (0.5 * cos(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45e-28], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2e+152], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.45000000000000006e-28

   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 66.8%

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

   if 1.45000000000000006e-28 < im < 2.0000000000000001e152

   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 81.5%

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

      1. *-commutative 81.5%

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

   4. Simplified 81.5%

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

   if 2.0000000000000001e152 < 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. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   4. Simplified 100.0%

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

   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%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 3: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.0095)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.0095) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.0095)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0095], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00949999999999999976

   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 86.7%

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

      1. +-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. fma-def 86.7%

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

   4. Simplified 86.7%

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

   if 0.00949999999999999976 < 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 78.6%

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

      1. *-commutative 78.6%

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

   4. Simplified 78.6%

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

   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. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   4. Simplified 100.0%

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

   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%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 4: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.45e-28) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.45e-28) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(-im) + exp(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 <= 1.45d-28) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.45e-28) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.45e-28:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.45e-28)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.45e-28)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.45e-28], N[Cos[re], $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 < 1.45000000000000006e-28

   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 66.8%

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

   if 1.45000000000000006e-28 < 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 81.8%

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

      1. *-commutative 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 5: 61.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9e+14)
   (cos re)
   (if (<= im 1.12e+145)
     (+ 0.25 (* 0.25 (pow re 2.0)))
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9e+14) {
		tmp = cos(re);
	} else if (im <= 1.12e+145) {
		tmp = 0.25 + (0.25 * pow(re, 2.0));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9e+14)
		tmp = cos(re);
	elseif (im <= 1.12e+145)
		tmp = Float64(0.25 + Float64(0.25 * (re ^ 2.0)));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9e+14], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.12e+145], N[(0.25 + N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 9e14

   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 66.3%

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

   if 9e14 < im < 1.12000000000000003e145

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.7%

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

      1. *-commutative 2.7%

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

   5. Simplified 2.7%

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

   6. Taylor expanded in re around 0 30.5%

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

      1. *-commutative 30.5%

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

   8. Simplified 30.5%

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

   if 1.12000000000000003e145 < 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 82.1%

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

      1. *-commutative 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in im around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. fma-def 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;0.25 + 0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 6: 61.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.06e+15)
   (cos re)
   (if (<= im 1.12e+145) (* 0.25 (pow re 2.0)) (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.06e+15) {
		tmp = cos(re);
	} else if (im <= 1.12e+145) {
		tmp = 0.25 * pow(re, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.06e+15)
		tmp = cos(re);
	elseif (im <= 1.12e+145)
		tmp = Float64(0.25 * (re ^ 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.06e+15], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.12e+145], N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.06e15

   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 66.3%

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

   if 1.06e15 < im < 1.12000000000000003e145

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.7%

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

      1. *-commutative 2.7%

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

   5. Simplified 2.7%

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

   6. Taylor expanded in re around 0 30.5%

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

      1. *-commutative 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in re around inf 29.9%

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

   if 1.12000000000000003e145 < 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 82.1%

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

      1. *-commutative 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in im around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. fma-def 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 7: 54.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {re}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.6e+14) (cos re) (* 0.25 (pow re 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.6e+14) {
		tmp = cos(re);
	} else {
		tmp = 0.25 * pow(re, 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 <= 9.6d+14) then
        tmp = cos(re)
    else
        tmp = 0.25d0 * (re ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 9.6e+14) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.25 * Math.pow(re, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.6e+14:
		tmp = math.cos(re)
	else:
		tmp = 0.25 * math.pow(re, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.6e+14)
		tmp = cos(re);
	else
		tmp = Float64(0.25 * (re ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 9.6e+14)
		tmp = cos(re);
	else
		tmp = 0.25 * (re ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.6e+14], N[Cos[re], $MachinePrecision], N[(0.25 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.6e14

   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 66.3%

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

   if 9.6e14 < im

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.7%

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

      1. *-commutative 2.7%

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

   5. Simplified 2.7%

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

   6. Taylor expanded in re around 0 20.3%

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

      1. *-commutative 20.3%

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

   8. Simplified 20.3%

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

   9. Taylor expanded in re around inf 19.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {re}^{2}\\ \end{array} \]

Alternative 8: 51.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (cos re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.cos(re)

Julia

function code(re, im)
	return cos(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Cos[re], $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 52.5%

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

3. Final simplification 52.5%

   \[\leadsto \cos re \]

Alternative 9: 3.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

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 78.0%

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

   1. +-commutative 78.0%

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

   2. unpow2 78.0%

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

   3. fma-def 78.0%

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

4. Simplified 78.0%

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

6. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-4 + \cos re} \]
7. Step-by-step derivation

   1. +-commutative 2.9%

      \[\leadsto \color{blue}{\cos re + -4} \]

8. Simplified 2.9%

   \[\leadsto \color{blue}{\cos re + -4} \]

9. Taylor expanded in re around 0 2.9%

   \[\leadsto \color{blue}{-3} \]

10. Final simplification 2.9%

    \[\leadsto -3 \]

Alternative 10: 8.2% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 8.7%

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

   1. *-commutative 8.7%

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

5. Simplified 8.7%

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

6. Taylor expanded in re around 0 8.8%

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

7. Final simplification 8.8%

   \[\leadsto 0.25 \]

Alternative 11: 29.1% 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 34.8%

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

   1. +-inverses 34.8%

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

   2. +-rgt-identity 34.8%

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

   3. *-inverses 34.8%

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

5. Simplified 34.8%

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

6. Final simplification 34.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))