math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 13

Speedup: 1.0×

• 49.3% 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, 0.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (cos re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	return cos(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

im = abs(im)
function code(re, im)
	return Float64(cos(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (* (cos re) 0.5) (+ (exp im) (exp (- im)))))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (cos(re) * 0.5d0) * (exp(im) + exp(-im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $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%

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

Alternative 3: 96.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 900:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 900.0)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 7.5e+76)
     (* 0.5 (log1p (expm1 (* im im))))
     (* 0.041666666666666664 (* (cos re) (pow im 4.0))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 900.0) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 7.5e+76) {
		tmp = 0.5 * log1p(expm1((im * im)));
	} else {
		tmp = 0.041666666666666664 * (cos(re) * pow(im, 4.0));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 900.0) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 7.5e+76) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * im)));
	} else {
		tmp = 0.041666666666666664 * (Math.cos(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 900.0:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 7.5e+76:
		tmp = 0.5 * math.log1p(math.expm1((im * im)))
	else:
		tmp = 0.041666666666666664 * (math.cos(re) * math.pow(im, 4.0))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 900.0)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 7.5e+76)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * im))));
	else
		tmp = Float64(0.041666666666666664 * Float64(cos(re) * (im ^ 4.0)));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 900.0], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.5e+76], N[(0.5 * N[Log[1 + N[(Exp[N[(im * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 900

   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 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

   if 900 < im < 7.4999999999999995e76

   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 3.9%

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

      1. unpow2 3.9%

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

   4. Simplified 3.9%

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

   5. Taylor expanded in im around inf 3.9%

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

      1. unpow2 3.9%

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

      2. associate-*r* 3.9%

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

      3. *-commutative 3.9%

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

      4. associate-*l* 3.9%

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

   7. Simplified 3.9%

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

   8. Taylor expanded in re around 0 3.3%

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

      1. pow2 3.3%

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

      2. log1p-expm1-u 84.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)} \]

   10. Applied egg-rr 84.2%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)} \]

   if 7.4999999999999995e76 < 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 98.2%

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

      1. unpow2 98.2%

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

      2. *-commutative 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 900:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+76}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\cos re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 92.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 900 \lor \neg \left(im \leq 2.6 \cdot 10^{+151}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (or (<= im 900.0) (not (<= im 2.6e+151)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (* 0.5 (log1p (expm1 (* im im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((im <= 900.0) || !(im <= 2.6e+151)) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * log1p(expm1((im * im)));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((im <= 900.0) || !(im <= 2.6e+151)) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * im)));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if (im <= 900.0) or not (im <= 2.6e+151):
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * im)))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if ((im <= 900.0) || !(im <= 2.6e+151))
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * im))));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[Or[LessEqual[im, 900.0], N[Not[LessEqual[im, 2.6e+151]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(im * im), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 900 or 2.60000000000000013e151 < 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 86.7%

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

      1. unpow2 86.7%

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

   4. Simplified 86.7%

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

   if 900 < im < 2.60000000000000013e151

   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 4.9%

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

      1. unpow2 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in im around inf 4.9%

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

      1. unpow2 4.9%

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

      2. associate-*r* 4.9%

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

      3. *-commutative 4.9%

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

      4. associate-*l* 4.9%

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

   7. Simplified 4.9%

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

   8. Taylor expanded in re around 0 3.6%

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

      1. pow2 3.6%

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

      2. log1p-expm1-u 74.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)} \]

   10. Applied egg-rr 74.3%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 900 \lor \neg \left(im \leq 2.6 \cdot 10^{+151}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 8.2 \cdot 10^{+42} \lor \neg \left(im \leq 2.6 \cdot 10^{+151}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{{im}^{6}}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (or (<= im 8.2e+42) (not (<= im 2.6e+151)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (* 0.5 (cbrt (pow im 6.0)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((im <= 8.2e+42) || !(im <= 2.6e+151)) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * cbrt(pow(im, 6.0));
	}
	return tmp;
}

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((im <= 8.2e+42) || !(im <= 2.6e+151)) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * Math.cbrt(Math.pow(im, 6.0));
	}
	return tmp;
}

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if ((im <= 8.2e+42) || !(im <= 2.6e+151))
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * cbrt((im ^ 6.0)));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[Or[LessEqual[im, 8.2e+42], N[Not[LessEqual[im, 2.6e+151]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[im, 6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.2000000000000001e42 or 2.60000000000000013e151 < 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 82.8%

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

      1. unpow2 82.8%

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

   4. Simplified 82.8%

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

   if 8.2000000000000001e42 < im < 2.60000000000000013e151

   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 5.6%

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

      1. unpow2 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in im around inf 5.6%

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

      1. unpow2 5.6%

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

      2. associate-*r* 5.6%

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

      3. *-commutative 5.6%

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

      4. associate-*l* 5.6%

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

   7. Simplified 5.6%

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

   8. Taylor expanded in re around 0 3.9%

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

      1. add-cbrt-cube 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{\left({im}^{2} \cdot {im}^{2}\right) \cdot {im}^{2}}}\right) \]

      2. pow2 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot {im}^{2}}\right) \]

      3. pow2 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot {im}^{2}}\right) \]

      4. pow2 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(im \cdot im\right)}}\right) \]

      5. pow3 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{\left(im \cdot im\right)}^{3}}}\right) \]

      6. pow2 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{\color{blue}{\left({im}^{2}\right)}}^{3}}\right) \]

      7. pow-pow 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 3\right)}}}\right) \]

      8. metadata-eval 62.8%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{im}^{\color{blue}{6}}}\right) \]

   10. Applied egg-rr 62.8%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{{im}^{6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.2 \cdot 10^{+42} \lor \neg \left(im \leq 2.6 \cdot 10^{+151}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{{im}^{6}}\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.5× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * (0.5d0 + (0.5d0 * exp(im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. +-commutative 100.0%

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

   4. distribute-lft-in 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-def 100.0%

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

   9. exp-neg 100.0%

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

   10. associate-*l/ 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 78.6%

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

   1. fma-udef 78.6%

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

6. Applied egg-rr 78.6%

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

7. Final simplification 78.6%

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

Alternative 7: 79.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 350:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 350.0)
   (cos re)
   (if (<= im 1.35e+154)
     (* (* im im) (+ 0.5 (* (* re re) -0.25)))
     (* (cos re) (* 0.5 (* im im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 350.0) {
		tmp = cos(re);
	} else if (im <= 1.35e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 350.0d0) then
        tmp = cos(re)
    else if (im <= 1.35d+154) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 350.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.35e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 350.0:
		tmp = math.cos(re)
	elif im <= 1.35e+154:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re) * (0.5 * (im * im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 350.0)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 350.0)
		tmp = cos(re);
	elseif (im <= 1.35e+154)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 350.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 350

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.0%

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

   5. Taylor expanded in im around 0 71.7%

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

   if 350 < 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 im around 0 5.1%

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

      1. unpow2 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around inf 5.1%

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

      1. unpow2 5.1%

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

      2. associate-*r* 5.1%

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

      3. *-commutative 5.1%

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

      4. associate-*l* 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in re around 0 24.8%

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

      1. associate-*r* 24.8%

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

      2. distribute-rgt-out 24.8%

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

      3. unpow2 24.8%

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

      4. +-commutative 24.8%

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

      5. *-commutative 24.8%

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

      6. unpow2 24.8%

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

   10. Simplified 24.8%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 350:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 80.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 390.0)
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1.35e+154)
     (* (* im im) (+ 0.5 (* (* re re) -0.25)))
     (* (cos re) (* 0.5 (* im im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 390.0) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 390.0d0) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = cos(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 390.0) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = Math.cos(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 390.0:
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	else:
		tmp = math.cos(re) * (0.5 * (im * im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 390.0)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = Float64(cos(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 390.0)
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	else
		tmp = cos(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 390.0], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 390

   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 85.5%

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

      1. unpow2 85.5%

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

   4. Simplified 85.5%

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

   if 390 < 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 im around 0 5.1%

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

      1. unpow2 5.1%

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

   4. Simplified 5.1%

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

   5. Taylor expanded in im around inf 5.1%

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

      1. unpow2 5.1%

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

      2. associate-*r* 5.1%

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

      3. *-commutative 5.1%

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

      4. associate-*l* 5.1%

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

   7. Simplified 5.1%

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

   8. Taylor expanded in re around 0 24.8%

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

      1. associate-*r* 24.8%

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

      2. distribute-rgt-out 24.8%

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

      3. unpow2 24.8%

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

      4. +-commutative 24.8%

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

      5. *-commutative 24.8%

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

      6. unpow2 24.8%

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

   10. Simplified 24.8%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 73.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 520.0) (cos re) (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = cos(re);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 520.0d0) then
        tmp = cos(re)
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 520.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 520.0:
		tmp = math.cos(re)
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 520.0)
		tmp = cos(re);
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 520.0)
		tmp = cos(re);
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 520.0], N[Cos[re], $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 520

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. exp-neg 100.0%

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

      10. associate-*l/ 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.0%

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

   5. Taylor expanded in im around 0 71.7%

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

   if 520 < 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 47.6%

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

      1. unpow2 47.6%

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

   4. Simplified 47.6%

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

   5. Taylor expanded in im around inf 47.6%

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

      1. unpow2 47.6%

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

      2. associate-*r* 47.6%

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

      3. *-commutative 47.6%

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

      4. associate-*l* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. associate-*r* 13.7%

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

      2. distribute-rgt-out 48.0%

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

      3. unpow2 48.0%

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

      4. +-commutative 48.0%

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

      5. *-commutative 48.0%

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

      6. unpow2 48.0%

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

   10. Simplified 48.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 10: 51.5% accurate, 23.6× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 125.0)
   (* 0.5 (+ 2.0 (* im im)))
   (* (* im im) (+ 0.5 (* (* re re) -0.25)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 125.0d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = (im * im) * (0.5d0 + ((re * re) * (-0.25d0)))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 125.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 125.0:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = (im * im) * (0.5 + ((re * re) * -0.25))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 125.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125.0)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = (im * im) * (0.5 + ((re * re) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 125.0], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 125

   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 85.5%

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

      1. unpow2 85.5%

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

   4. Simplified 85.5%

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

   5. Taylor expanded in re around 0 44.6%

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

      1. add-cbrt-cube 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{\left({im}^{2} \cdot {im}^{2}\right) \cdot {im}^{2}}}\right) \]

      2. pow2 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot {im}^{2}}\right) \]

      3. pow2 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot {im}^{2}}\right) \]

      4. pow2 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(im \cdot im\right)}}\right) \]

      5. pow3 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{\left(im \cdot im\right)}^{3}}}\right) \]

      6. pow2 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{\color{blue}{\left({im}^{2}\right)}}^{3}}\right) \]

      7. pow-pow 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 3\right)}}}\right) \]

      8. metadata-eval 52.5%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{im}^{\color{blue}{6}}}\right) \]

   7. Applied egg-rr 52.5%

      \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{{im}^{6}}}\right) \]

   8. Taylor expanded in im around 0 44.6%

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

      1. unpow2 44.6%

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

   10. Simplified 44.6%

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

   if 125 < 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 47.6%

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

      1. unpow2 47.6%

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

   4. Simplified 47.6%

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

   5. Taylor expanded in im around inf 47.6%

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

      1. unpow2 47.6%

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

      2. associate-*r* 47.6%

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

      3. *-commutative 47.6%

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

      4. associate-*l* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in re around 0 13.7%

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

      1. associate-*r* 13.7%

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

      2. distribute-rgt-out 48.0%

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

      3. unpow2 48.0%

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

      4. +-commutative 48.0%

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

      5. *-commutative 48.0%

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

      6. unpow2 48.0%

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

   10. Simplified 48.0%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25 + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 11: 48.5% accurate, 27.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= re 8.2e+16)
   (* 0.5 (+ 2.0 (* im im)))
   (* -0.25 (* (* re im) (* re im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (re <= 8.2e+16) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = -0.25 * ((re * im) * (re * im));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 8.2d+16) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = (-0.25d0) * ((re * im) * (re * im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (re <= 8.2e+16) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = -0.25 * ((re * im) * (re * im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if re <= 8.2e+16:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = -0.25 * ((re * im) * (re * im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (re <= 8.2e+16)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(-0.25 * Float64(Float64(re * im) * Float64(re * im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 8.2e+16)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = -0.25 * ((re * im) * (re * im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[re, 8.2e+16], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(re * im), $MachinePrecision] * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 8.2e16

   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 71.4%

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

      1. unpow2 71.4%

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

   4. Simplified 71.4%

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

   5. Taylor expanded in re around 0 46.5%

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

      1. add-cbrt-cube 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{\left({im}^{2} \cdot {im}^{2}\right) \cdot {im}^{2}}}\right) \]

      2. pow2 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot {im}^{2}}\right) \]

      3. pow2 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot {im}^{2}}\right) \]

      4. pow2 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(im \cdot im\right)}}\right) \]

      5. pow3 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{\left(im \cdot im\right)}^{3}}}\right) \]

      6. pow2 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{\color{blue}{\left({im}^{2}\right)}}^{3}}\right) \]

      7. pow-pow 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 3\right)}}}\right) \]

      8. metadata-eval 60.7%

         \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{im}^{\color{blue}{6}}}\right) \]

   7. Applied egg-rr 60.7%

      \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{{im}^{6}}}\right) \]

   8. Taylor expanded in im around 0 46.5%

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

      1. unpow2 46.5%

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

   10. Simplified 46.5%

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

   if 8.2e16 < re

   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 88.1%

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

      1. unpow2 88.1%

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

   4. Simplified 88.1%

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

   5. Taylor expanded in im around inf 28.1%

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

      1. unpow2 28.1%

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

      2. associate-*r* 28.1%

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

      3. *-commutative 28.1%

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

      4. associate-*l* 28.1%

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

   7. Simplified 28.1%

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

   8. Taylor expanded in re around 0 7.6%

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

      1. associate-*r* 7.6%

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

      2. distribute-rgt-out 18.5%

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

      3. unpow2 18.5%

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

      4. +-commutative 18.5%

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

      5. *-commutative 18.5%

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

      6. unpow2 18.5%

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

   10. Simplified 18.5%

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

   11. Taylor expanded in re around inf 18.5%

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

       1. unpow2 18.5%

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

       2. unpow2 18.5%

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

       3. *-commutative 18.5%

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

       4. unswap-sqr 19.1%

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

   13. Simplified 19.1%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(re \cdot im\right) \cdot \left(re \cdot im\right)\right)\\ \end{array} \]

Alternative 12: 47.1% accurate, 44.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 0.5 \cdot \left(2 + im \cdot im\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* 0.5 (+ 2.0 (* im im))))

C

im = abs(im);
double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (2.0d0 + (im * im))
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 0.5 * (2.0 + (im * im));
}

Python

im = abs(im)
def code(re, im):
	return 0.5 * (2.0 + (im * im))

Julia

im = abs(im)
function code(re, im)
	return Float64(0.5 * Float64(2.0 + Float64(im * im)))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 0.5 * (2.0 + (im * im));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(0.5 * N[(2.0 + N[(im * 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. Taylor expanded in im around 0 75.5%

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

   1. unpow2 75.5%

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

4. Simplified 75.5%

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

5. Taylor expanded in re around 0 40.0%

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

   1. add-cbrt-cube 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{\left({im}^{2} \cdot {im}^{2}\right) \cdot {im}^{2}}}\right) \]

   2. pow2 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right) \cdot {im}^{2}}\right) \]

   3. pow2 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right) \cdot {im}^{2}}\right) \]

   4. pow2 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(im \cdot im\right)}}\right) \]

   5. pow3 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{\left(im \cdot im\right)}^{3}}}\right) \]

   6. pow2 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{\color{blue}{\left({im}^{2}\right)}}^{3}}\right) \]

   7. pow-pow 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{\color{blue}{{im}^{\left(2 \cdot 3\right)}}}\right) \]

   8. metadata-eval 51.4%

      \[\leadsto 0.5 \cdot \left(2 + \sqrt[3]{{im}^{\color{blue}{6}}}\right) \]

7. Applied egg-rr 51.4%

   \[\leadsto 0.5 \cdot \left(2 + \color{blue}{\sqrt[3]{{im}^{6}}}\right) \]

8. Taylor expanded in im around 0 40.0%

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

   1. unpow2 40.0%

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

10. Simplified 40.0%

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

11. Final simplification 40.0%

    \[\leadsto 0.5 \cdot \left(2 + im \cdot im\right) \]

Alternative 13: 21.4% accurate, 61.6× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* 0.5 (* im im)))

C

im = abs(im);
double code(double re, double im) {
	return 0.5 * (im * im);
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * im)
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 0.5 * (im * im);
}

Python

im = abs(im)
def code(re, im):
	return 0.5 * (im * im)

Julia

im = abs(im)
function code(re, im)
	return Float64(0.5 * Float64(im * im))
end

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(0.5 * N[(im * im), $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 75.5%

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

   1. unpow2 75.5%

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

4. Simplified 75.5%

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

5. Taylor expanded in im around inf 25.4%

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

   1. unpow2 25.4%

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

   2. associate-*r* 25.4%

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

   3. *-commutative 25.4%

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

   4. associate-*l* 25.4%

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

7. Simplified 25.4%

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

8. Taylor expanded in re around 0 16.2%

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

   1. unpow2 16.2%

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

10. Simplified 16.2%

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

11. Final simplification 16.2%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))