math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 11

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.12e-7)
   (cos re)
   (if (<= im 6e+151)
     (* (fma im im 2.0) (+ 0.5 (* -0.25 (pow re 2.0))))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = cos(re);
	} else if (im <= 6e+151) {
		tmp = fma(im, im, 2.0) * (0.5 + (-0.25 * pow(re, 2.0)));
	} else {
		tmp = (0.5 * cos(re)) * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 6e+151)
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 + Float64(-0.25 * (re ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 6e+151], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[Power[re, 2.0], $MachinePrecision]), $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 < 1.12e-7

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-def 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.12e-7 < im < 5.9999999999999998e151

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 17.8%

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

      1. +-commutative 17.8%

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

      2. unpow2 17.8%

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

      3. fma-def 17.8%

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

   4. Simplified 17.8%

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

   5. Taylor expanded in re around 0 31.1%

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

      1. associate-*r* 31.1%

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

      2. distribute-rgt-out 31.1%

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

      3. +-commutative 31.1%

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

      4. unpow2 31.1%

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

      5. fma-udef 31.1%

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

      6. +-commutative 31.1%

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

   7. Simplified 31.1%

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

   if 5.9999999999999998e151 < 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 97.8%

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

      1. +-commutative 97.8%

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

      2. unpow2 97.8%

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

      3. fma-def 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in im around inf 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 97.8%

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

      3. *-commutative 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 3: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.12e-7)
   (cos re)
   (if (<= im 1.02e+142)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (* (* 0.5 (cos re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = cos(re);
	} else if (im <= 1.02e+142) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} 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 <= 1.12d-7) then
        tmp = cos(re)
    else if (im <= 1.02d+142) then
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    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 <= 1.12e-7) {
		tmp = Math.cos(re);
	} else if (im <= 1.02e+142) {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	} else {
		tmp = (0.5 * Math.cos(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.12e-7:
		tmp = math.cos(re)
	elif im <= 1.02e+142:
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	else:
		tmp = (0.5 * math.cos(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 1.02e+142)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	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 <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 1.02e+142)
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	else
		tmp = (0.5 * cos(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.02e+142], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $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 < 1.12e-7

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-def 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.12e-7 < im < 1.0199999999999999e142

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 18.2%

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

      1. +-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. fma-def 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in re around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. +-commutative 32.8%

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

      4. unpow2 32.8%

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

      5. fma-udef 32.8%

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

      6. +-commutative 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in im around 0 18.8%

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

      1. distribute-rgt-in 18.8%

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

      2. metadata-eval 18.8%

         \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

      3. *-commutative 18.8%

         \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

      4. associate-*l* 18.8%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

      5. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   if 1.0199999999999999e142 < 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 95.7%

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

      1. +-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. fma-def 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in im around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

   7. Simplified 95.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 4: 60.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(im \cdot \sqrt{0.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.12e-7)
   (cos re)
   (if (<= im 1.12e+142)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (pow (* im (sqrt 0.5)) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = cos(re);
	} else if (im <= 1.12e+142) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} else {
		tmp = pow((im * sqrt(0.5)), 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 <= 1.12d-7) then
        tmp = cos(re)
    else if (im <= 1.12d+142) then
        tmp = 1.0d0 + ((re ** 2.0d0) * (-0.5d0))
    else
        tmp = (im * sqrt(0.5d0)) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = Math.cos(re);
	} else if (im <= 1.12e+142) {
		tmp = 1.0 + (Math.pow(re, 2.0) * -0.5);
	} else {
		tmp = Math.pow((im * Math.sqrt(0.5)), 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.12e-7:
		tmp = math.cos(re)
	elif im <= 1.12e+142:
		tmp = 1.0 + (math.pow(re, 2.0) * -0.5)
	else:
		tmp = math.pow((im * math.sqrt(0.5)), 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 1.12e+142)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	else
		tmp = Float64(im * sqrt(0.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 1.12e+142)
		tmp = 1.0 + ((re ^ 2.0) * -0.5);
	else
		tmp = (im * sqrt(0.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.12e+142], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[Power[N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.12e-7

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-def 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.12e-7 < im < 1.11999999999999996e142

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 18.2%

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

      1. +-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. fma-def 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in re around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. +-commutative 32.8%

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

      4. unpow2 32.8%

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

      5. fma-udef 32.8%

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

      6. +-commutative 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in im around 0 18.8%

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

      1. distribute-rgt-in 18.8%

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

      2. metadata-eval 18.8%

         \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

      3. *-commutative 18.8%

         \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

      4. associate-*l* 18.8%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

      5. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   if 1.11999999999999996e142 < 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 95.7%

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

      1. +-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. fma-def 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in im around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*l* 95.7%

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

      3. *-commutative 95.7%

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

   7. Simplified 95.7%

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

      1. add-sqr-sqrt 73.7%

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

      2. pow2 73.7%

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

      3. sqrt-prod 73.7%

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

      4. unpow2 73.7%

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

      5. sqrt-prod 73.7%

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

      6. add-sqr-sqrt 73.7%

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

   9. Applied egg-rr 73.7%

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

   10. Taylor expanded in re around 0 73.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(im \cdot \sqrt{0.5}\right)}^{2}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (cos re)) (fma im im 2.0)))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * fma(im, im, 2.0);
}

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * fma(im, im, 2.0))
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $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 82.2%

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

   1. +-commutative 82.2%

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

   2. unpow2 82.2%

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

   3. fma-def 82.2%

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

4. Simplified 82.2%

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

5. Final simplification 82.2%

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

Alternative 6: 60.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.12e-7)
   (cos re)
   (if (<= im 1.12e+142)
     (+ 1.0 (* (pow re 2.0) -0.5))
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = cos(re);
	} else if (im <= 1.12e+142) {
		tmp = 1.0 + (pow(re, 2.0) * -0.5);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e-7)
		tmp = cos(re);
	elseif (im <= 1.12e+142)
		tmp = Float64(1.0 + Float64((re ^ 2.0) * -0.5));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e-7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.12e+142], N[(1.0 + N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.12e-7

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-def 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.12e-7 < im < 1.11999999999999996e142

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 18.2%

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

      1. +-commutative 18.2%

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

      2. unpow2 18.2%

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

      3. fma-def 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in re around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. distribute-rgt-out 32.8%

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

      3. +-commutative 32.8%

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

      4. unpow2 32.8%

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

      5. fma-udef 32.8%

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

      6. +-commutative 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in im around 0 18.8%

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

      1. distribute-rgt-in 18.8%

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

      2. metadata-eval 18.8%

         \[\leadsto \color{blue}{1} + \left(-0.25 \cdot {re}^{2}\right) \cdot 2 \]

      3. *-commutative 18.8%

         \[\leadsto 1 + \color{blue}{\left({re}^{2} \cdot -0.25\right)} \cdot 2 \]

      4. associate-*l* 18.8%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot 2\right)} \]

      5. metadata-eval 18.8%

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

   10. Simplified 18.8%

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

   if 1.11999999999999996e142 < 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 95.7%

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

      1. +-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. fma-def 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in re around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. unpow2 73.7%

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

      3. fma-udef 73.7%

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

   7. Simplified 73.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+142}:\\ \;\;\;\;1 + {re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 7: 60.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+142}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 680.0)
   (cos re)
   (if (<= im 1.1e+142) (* (pow re 2.0) -0.5) (* 0.5 (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = cos(re);
	} else if (im <= 1.1e+142) {
		tmp = pow(re, 2.0) * -0.5;
	} 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 <= 680.0d0) then
        tmp = cos(re)
    else if (im <= 1.1d+142) then
        tmp = (re ** 2.0d0) * (-0.5d0)
    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 <= 680.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.1e+142) {
		tmp = Math.pow(re, 2.0) * -0.5;
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 680.0:
		tmp = math.cos(re)
	elif im <= 1.1e+142:
		tmp = math.pow(re, 2.0) * -0.5
	else:
		tmp = 0.5 * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 680.0)
		tmp = cos(re);
	elseif (im <= 1.1e+142)
		tmp = Float64((re ^ 2.0) * -0.5);
	else
		tmp = Float64(0.5 * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 680.0)
		tmp = cos(re);
	elseif (im <= 1.1e+142)
		tmp = (re ^ 2.0) * -0.5;
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 680.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.1e+142], N[(N[Power[re, 2.0], $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 680

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. unpow2 84.7%

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

      3. fma-def 84.7%

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

   4. Simplified 84.7%

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

   5. Taylor expanded in im around 0 67.2%

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

   if 680 < im < 1.09999999999999993e142

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.3%

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

      1. +-commutative 5.3%

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

      2. unpow2 5.3%

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

      3. fma-def 5.3%

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

   4. Simplified 5.3%

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

   5. Taylor expanded in re around 0 24.0%

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

      1. associate-*r* 24.0%

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

      2. distribute-rgt-out 24.0%

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

      3. +-commutative 24.0%

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

      4. unpow2 24.0%

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

      5. fma-udef 24.0%

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

      6. +-commutative 24.0%

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

   7. Simplified 24.0%

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

   8. Taylor expanded in re around inf 22.5%

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

      1. associate-*r* 22.5%

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

      2. +-commutative 22.5%

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

      3. unpow2 22.5%

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

      4. fma-udef 22.5%

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

      5. *-commutative 22.5%

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

      6. associate-*r* 22.5%

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

   10. Simplified 22.5%

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

   11. Taylor expanded in im around 0 8.9%

       \[\leadsto \color{blue}{-0.5 \cdot {re}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative 8.9%

          \[\leadsto \color{blue}{{re}^{2} \cdot -0.5} \]

   13. Simplified 8.9%

       \[\leadsto \color{blue}{{re}^{2} \cdot -0.5} \]

   if 1.09999999999999993e142 < 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 95.7%

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

      1. +-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. fma-def 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in re around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. unpow2 73.7%

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

      3. fma-udef 73.7%

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

   7. Simplified 73.7%

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

   8. Taylor expanded in im around inf 73.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+142}:\\ \;\;\;\;{re}^{2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]

Alternative 8: 59.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.12e-7) (cos re) (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.12e-7) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.12e-7)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.12e-7], N[Cos[re], $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.12e-7

   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 85.2%

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

      1. +-commutative 85.2%

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

      2. unpow2 85.2%

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

      3. fma-def 85.2%

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

   4. Simplified 85.2%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.12e-7 < im

   1. Initial program 99.9%

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

   2. Taylor expanded in im around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. unpow2 72.0%

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

      3. fma-def 72.0%

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

   4. Simplified 72.0%

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

   5. Taylor expanded in re around 0 56.4%

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

      1. +-commutative 56.4%

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

      2. unpow2 56.4%

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

      3. fma-udef 56.4%

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

   7. Simplified 56.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 9: 59.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+85}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.22e+85) (cos re) (* 0.5 (pow im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+85) {
		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 <= 1.22d+85) 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 <= 1.22e+85) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.22e+85:
		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 <= 1.22e+85)
		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 <= 1.22e+85)
		tmp = cos(re);
	else
		tmp = 0.5 * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.22e+85], N[Cos[re], $MachinePrecision], N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e85

   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 82.4%

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

      1. +-commutative 82.4%

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

      2. unpow2 82.4%

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

      3. fma-def 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in im around 0 65.4%

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

   if 1.22e85 < 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 81.1%

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

      1. +-commutative 81.1%

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

      2. unpow2 81.1%

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

      3. fma-def 81.1%

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

   4. Simplified 81.1%

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

   5. Taylor expanded in re around 0 62.4%

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

      1. +-commutative 62.4%

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

      2. unpow2 62.4%

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

      3. fma-udef 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in im around inf 62.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+85}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {im}^{2}\\ \end{array} \]

Alternative 10: 51.2% 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 82.2%

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

   1. +-commutative 82.2%

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

   2. unpow2 82.2%

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

   3. fma-def 82.2%

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

4. Simplified 82.2%

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

5. Taylor expanded in im around 0 53.4%

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

6. Final simplification 53.4%

   \[\leadsto \cos re \]

Alternative 11: 28.9% 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. Taylor expanded in im around 0 82.2%

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

   1. +-commutative 82.2%

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

   2. unpow2 82.2%

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

   3. fma-def 82.2%

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

4. Simplified 82.2%

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

5. Taylor expanded in re around 0 49.4%

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

   1. +-commutative 49.4%

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

   2. unpow2 49.4%

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

   3. fma-udef 49.4%

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

7. Simplified 49.4%

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

8. Taylor expanded in im around 0 28.1%

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

9. Final simplification 28.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))