math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 10

Speedup: 1.0×

• 50.1% 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: 71.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e-21)
   (cos re)
   (if (<= im 1.35e+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 <= 2.8e-21) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * (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 <= 2.8d-21) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.8e-21:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.8e-21)
		tmp = cos(re);
	elseif (im <= 1.35e+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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.8e-21)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e-21], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+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 < 2.80000000000000004e-21

   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 71.4%

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

   if 2.80000000000000004e-21 < im < 1.35000000000000003e154

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 66.0%

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

   if 1.35000000000000003e154 < 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%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {im}^{2}\\ \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow im 2.0))))
   (if (<= im 1.4)
     (* (cos re) (+ t_0 1.0))
     (if (<= im 1.35e+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 <= 1.4) {
		tmp = cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+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 <= 1.4d0) then
        tmp = cos(re) * (t_0 + 1.0d0)
    else if (im <= 1.35d+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 <= 1.4) {
		tmp = Math.cos(re) * (t_0 + 1.0);
	} else if (im <= 1.35e+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 <= 1.4:
		tmp = math.cos(re) * (t_0 + 1.0)
	elif im <= 1.35e+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 <= 1.4)
		tmp = Float64(cos(re) * Float64(t_0 + 1.0));
	elseif (im <= 1.35e+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 <= 1.4)
		tmp = cos(re) * (t_0 + 1.0);
	elseif (im <= 1.35e+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, 1.4], N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+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 < 1.3999999999999999

   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 86.4%

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

   5. Simplified 86.4%

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

   if 1.3999999999999999 < im < 1.35000000000000003e154

   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 65.3%

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

   if 1.35000000000000003e154 < 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%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.8e-21) (cos re) (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.8e-21) {
		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 <= 2.8d-21) 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 <= 2.8e-21) {
		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 <= 2.8e-21:
		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 <= 2.8e-21)
		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 <= 2.8e-21)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.8e-21], 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 < 2.80000000000000004e-21

   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 71.4%

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

   if 2.80000000000000004e-21 < 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 67.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{-21}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.08 \cdot 10^{+43}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+155}:\\ \;\;\;\;-0.25 \cdot {\left(re \cdot im\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.08e+43)
   (cos re)
   (if (<= im 5e+155) (* -0.25 (pow (* re im) 2.0)) (fma (* 0.5 im) im 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.08e+43) {
		tmp = cos(re);
	} else if (im <= 5e+155) {
		tmp = -0.25 * pow((re * im), 2.0);
	} else {
		tmp = fma((0.5 * im), im, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.08e+43)
		tmp = cos(re);
	elseif (im <= 5e+155)
		tmp = Float64(-0.25 * (Float64(re * im) ^ 2.0));
	else
		tmp = fma(Float64(0.5 * im), im, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.08e+43], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5e+155], N[(-0.25 * N[Power[N[(re * im), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.08e43

   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 67.3%

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

   if 1.08e43 < im < 4.9999999999999999e155

   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 9.7%

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

   5. Simplified 9.7%

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

   6. Taylor expanded in im around inf 9.7%

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

   7. Taylor expanded in re around 0 37.3%

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

      1. +-commutative 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-*r* 37.3%

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

      4. distribute-rgt-out 41.2%

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

   9. Simplified 41.2%

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

   10. Taylor expanded in re around inf 39.2%

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

       1. unpow2 39.2%

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

       2. unpow2 39.2%

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

       3. swap-sqr 39.2%

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

       4. unpow2 39.2%

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

   12. Simplified 39.2%

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

   if 4.9999999999999999e155 < 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 re around 0 71.9%

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

      1. +-commutative 71.9%

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

      2. unpow2 71.9%

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

      3. associate-*r* 71.9%

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

      4. fma-def 71.9%

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

   8. Applied egg-rr 71.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.08 \cdot 10^{+43}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+155}:\\ \;\;\;\;-0.25 \cdot {\left(re \cdot im\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.6e+66) (cos re) (fma (* 0.5 im) im 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.6e+66) {
		tmp = cos(re);
	} else {
		tmp = fma((0.5 * im), im, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.6e+66)
		tmp = cos(re);
	else
		tmp = fma(Float64(0.5 * im), im, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.6e+66], N[Cos[re], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.6e66

   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 64.8%

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

   if 4.6e66 < 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 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in re around 0 47.6%

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

      1. +-commutative 47.6%

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

      2. unpow2 47.6%

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

      3. associate-*r* 47.6%

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

      4. fma-def 47.6%

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

   8. Applied egg-rr 47.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e+69) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+69) {
		tmp = cos(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 <= 3.6d+69) then
        tmp = cos(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 <= 3.6e+69) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+69:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+69)
		tmp = cos(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 <= 3.6e+69)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+69], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.6000000000000003e69

   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 64.8%

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

   if 3.6000000000000003e69 < 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 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in im around inf 68.4%

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

   7. Taylor expanded in re around 0 47.6%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.4% 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. Add Preprocessing

3. Taylor expanded in im around 0 52.8%

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

4. Final simplification 52.8%

   \[\leadsto \cos re \]
5. Add Preprocessing

Alternative 9: 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 8.0%

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

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

Alternative 10: 28.2% 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%

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

3. Taylor expanded in im around 0 76.4%

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

5. Simplified 76.4%

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

6. Taylor expanded in re around 0 44.8%

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

7. Taylor expanded in im around 0 27.0%

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

8. Final simplification 27.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))