math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 11

Speedup: 1.0×

• 49.2% 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: 93.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 14000 \lor \neg \left(im \leq 2 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (or (<= im 14000.0) (not (<= im 2e+154)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (* 0.5 (+ (exp im) (exp (- im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((im <= 14000.0) || !(im <= 2e+154)) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (exp(im) + exp(-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 <= 14000.0d0) .or. (.not. (im <= 2d+154))) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * (exp(im) + exp(-im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((im <= 14000.0) || !(im <= 2e+154)) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (Math.exp(im) + Math.exp(-im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if (im <= 14000.0) or not (im <= 2e+154):
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = 0.5 * (math.exp(im) + math.exp(-im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if ((im <= 14000.0) || !(im <= 2e+154))
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(exp(im) + exp(Float64(-im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 14000.0) || ~((im <= 2e+154)))
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = 0.5 * (exp(im) + exp(-im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[Or[LessEqual[im, 14000.0], N[Not[LessEqual[im, 2e+154]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 14000 or 2.00000000000000007e154 < 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 87.0%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if 14000 < im < 2.00000000000000007e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 83.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14000 \lor \neg \left(im \leq 2 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + e^{-im}\right)\\ \end{array} \]

Alternative 4: 98.9% 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 77.0%

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

5. Taylor expanded in re around inf 77.0%

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

6. Final simplification 77.0%

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

Alternative 5: 93.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 14000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (or (<= im 14000.0) (not (<= im 1.35e+154)))
   (* (* (cos re) 0.5) (+ 2.0 (* im im)))
   (+ 0.5 (* 0.5 (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if ((im <= 14000.0) || !(im <= 1.35e+154)) {
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (0.5 * exp(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 <= 14000.0d0) .or. (.not. (im <= 1.35d+154))) then
        tmp = (cos(re) * 0.5d0) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if ((im <= 14000.0) || !(im <= 1.35e+154)) {
		tmp = (Math.cos(re) * 0.5) * (2.0 + (im * im));
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if (im <= 14000.0) or not (im <= 1.35e+154):
		tmp = (math.cos(re) * 0.5) * (2.0 + (im * im))
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if ((im <= 14000.0) || !(im <= 1.35e+154))
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 14000.0) || ~((im <= 1.35e+154)))
		tmp = (cos(re) * 0.5) * (2.0 + (im * im));
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[Or[LessEqual[im, 14000.0], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 14000 or 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 87.0%

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

      1. unpow2 87.0%

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

   4. Simplified 87.0%

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

   if 14000 < im < 1.35000000000000003e154

   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 0 100.0%

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

   5. Taylor expanded in re around 0 83.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 6: 87.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 14000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 14000.0) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 14000.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 * exp(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 <= 14000.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 14000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 14000.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 14000.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 14000.0)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 14000.0], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 14000

   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 69.8%

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

   if 14000 < im

   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 0 100.0%

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

   5. Taylor expanded in re around 0 79.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \]

Alternative 7: 78.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 160000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 160000000.0)
   (cos re)
   (if (<= im 2.7e+154)
     (+
      1.0
      (/
       (* (* im im) (- 0.25 (* (* im im) 0.0625)))
       (* im (- 0.5 (* im 0.25)))))
     (+ 1.0 (* im (* im 0.25))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 160000000.0) {
		tmp = cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	} else {
		tmp = 1.0 + (im * (im * 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 <= 160000000.0d0) then
        tmp = cos(re)
    else if (im <= 2.7d+154) then
        tmp = 1.0d0 + (((im * im) * (0.25d0 - ((im * im) * 0.0625d0))) / (im * (0.5d0 - (im * 0.25d0))))
    else
        tmp = 1.0d0 + (im * (im * 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 <= 160000000.0) {
		tmp = Math.cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	} else {
		tmp = 1.0 + (im * (im * 0.25));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 160000000.0:
		tmp = math.cos(re)
	elif im <= 2.7e+154:
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))))
	else:
		tmp = 1.0 + (im * (im * 0.25))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 160000000.0)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.25 - Float64(Float64(im * im) * 0.0625))) / Float64(im * Float64(0.5 - Float64(im * 0.25)))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * 0.25)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 160000000.0)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	else
		tmp = 1.0 + (im * (im * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 160000000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.25 - N[(N[(im * im), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im * N[(0.5 - N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.6e8

   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 69.8%

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

   if 1.6e8 < im < 2.70000000000000006e154

   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 0 100.0%

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

   5. Taylor expanded in re around 0 83.3%

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

   6. Taylor expanded in im around 0 3.8%

      \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
   7. Step-by-step derivation

      1. +-commutative 3.8%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      2. unpow2 3.8%

         \[\leadsto 1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      3. associate-*r* 3.8%

         \[\leadsto 1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right) \]

      4. distribute-rgt-out 3.8%

         \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   8. Simplified 3.8%

      \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 3.8%

         \[\leadsto 1 + \color{blue}{\left(im \cdot 0.5 + im \cdot \left(0.25 \cdot im\right)\right)} \]

      2. *-commutative 3.8%

         \[\leadsto 1 + \left(\color{blue}{0.5 \cdot im} + im \cdot \left(0.25 \cdot im\right)\right) \]

      3. flip-+ 27.2%

         \[\leadsto 1 + \color{blue}{\frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)}} \]

      4. *-commutative 27.2%

         \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot 0.5\right)} \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      5. *-commutative 27.2%

         \[\leadsto 1 + \frac{\left(im \cdot 0.5\right) \cdot \color{blue}{\left(im \cdot 0.5\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      6. swap-sqr 27.2%

         \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot 0.5\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      7. metadata-eval 27.2%

         \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \color{blue}{0.25} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      8. associate-*r* 27.2%

         \[\leadsto 1 + \frac{\color{blue}{im \cdot \left(im \cdot 0.25\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      9. *-commutative 27.2%

         \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \color{blue}{\left(im \cdot 0.25\right)}\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      10. *-commutative 27.2%

          \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      11. *-commutative 27.2%

          \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \color{blue}{\left(im \cdot 0.25\right)}} \]

   10. Applied egg-rr 27.2%

       \[\leadsto 1 + \color{blue}{\frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* 27.2%

          \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot 0.25} - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       2. swap-sqr 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot 0.25 - \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot 0.25\right) \cdot \left(im \cdot 0.25\right)\right)}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       3. unpow2 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot 0.25 - \left(im \cdot im\right) \cdot \color{blue}{{\left(im \cdot 0.25\right)}^{2}}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       4. distribute-lft-out-- 27.2%

          \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.25 - {\left(im \cdot 0.25\right)}^{2}\right)}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       5. unpow2 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \color{blue}{\left(im \cdot 0.25\right) \cdot \left(im \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       6. swap-sqr 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \color{blue}{\left(im \cdot im\right) \cdot \left(0.25 \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       7. metadata-eval 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot \color{blue}{0.0625}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       8. *-commutative 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{\color{blue}{im \cdot 0.5} - im \cdot \left(im \cdot 0.25\right)} \]

       9. distribute-lft-out-- 27.2%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{\color{blue}{im \cdot \left(0.5 - im \cdot 0.25\right)}} \]

   12. Simplified 27.2%

       \[\leadsto 1 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}} \]

   if 2.70000000000000006e154 < im

   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 0 100.0%

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

   5. Taylor expanded in re around 0 76.9%

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

   6. Taylor expanded in im around 0 76.9%

      \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
   7. Step-by-step derivation

      1. +-commutative 76.9%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      2. unpow2 76.9%

         \[\leadsto 1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      3. associate-*r* 76.9%

         \[\leadsto 1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right) \]

      4. distribute-rgt-out 76.9%

         \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   8. Simplified 76.9%

      \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   9. Taylor expanded in im around inf 76.9%

      \[\leadsto 1 + \color{blue}{0.25 \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 76.9%

          \[\leadsto 1 + 0.25 \cdot \color{blue}{\left(im \cdot im\right)} \]

       2. *-commutative 76.9%

          \[\leadsto 1 + \color{blue}{\left(im \cdot im\right) \cdot 0.25} \]

       3. associate-*r* 76.9%

          \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]

   11. Simplified 76.9%

       \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 160000000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \]

Alternative 8: 56.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 2.7e+154)
   (+
    1.0
    (/ (* (* im im) (- 0.25 (* (* im im) 0.0625))) (* im (- 0.5 (* im 0.25)))))
   (+ 1.0 (* im (* im 0.25)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 2.7e+154) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	} else {
		tmp = 1.0 + (im * (im * 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 <= 2.7d+154) then
        tmp = 1.0d0 + (((im * im) * (0.25d0 - ((im * im) * 0.0625d0))) / (im * (0.5d0 - (im * 0.25d0))))
    else
        tmp = 1.0d0 + (im * (im * 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 <= 2.7e+154) {
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	} else {
		tmp = 1.0 + (im * (im * 0.25));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 2.7e+154:
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))))
	else:
		tmp = 1.0 + (im * (im * 0.25))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 2.7e+154)
		tmp = Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(0.25 - Float64(Float64(im * im) * 0.0625))) / Float64(im * Float64(0.5 - Float64(im * 0.25)))));
	else
		tmp = Float64(1.0 + Float64(im * Float64(im * 0.25)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.7e+154)
		tmp = 1.0 + (((im * im) * (0.25 - ((im * im) * 0.0625))) / (im * (0.5 - (im * 0.25))));
	else
		tmp = 1.0 + (im * (im * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 2.7e+154], N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(0.25 - N[(N[(im * im), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im * N[(0.5 - N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.70000000000000006e154

   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 72.8%

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

   5. Taylor expanded in re around 0 45.5%

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

   6. Taylor expanded in im around 0 47.3%

      \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
   7. Step-by-step derivation

      1. +-commutative 47.3%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      2. unpow2 47.3%

         \[\leadsto 1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      3. associate-*r* 47.3%

         \[\leadsto 1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right) \]

      4. distribute-rgt-out 47.3%

         \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   8. Simplified 47.3%

      \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 47.3%

         \[\leadsto 1 + \color{blue}{\left(im \cdot 0.5 + im \cdot \left(0.25 \cdot im\right)\right)} \]

      2. *-commutative 47.3%

         \[\leadsto 1 + \left(\color{blue}{0.5 \cdot im} + im \cdot \left(0.25 \cdot im\right)\right) \]

      3. flip-+ 44.8%

         \[\leadsto 1 + \color{blue}{\frac{\left(0.5 \cdot im\right) \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)}} \]

      4. *-commutative 44.8%

         \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot 0.5\right)} \cdot \left(0.5 \cdot im\right) - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      5. *-commutative 44.8%

         \[\leadsto 1 + \frac{\left(im \cdot 0.5\right) \cdot \color{blue}{\left(im \cdot 0.5\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      6. swap-sqr 44.8%

         \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot 0.5\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      7. metadata-eval 44.8%

         \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \color{blue}{0.25} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      8. associate-*r* 44.8%

         \[\leadsto 1 + \frac{\color{blue}{im \cdot \left(im \cdot 0.25\right)} - \left(im \cdot \left(0.25 \cdot im\right)\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      9. *-commutative 44.8%

         \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \color{blue}{\left(im \cdot 0.25\right)}\right) \cdot \left(im \cdot \left(0.25 \cdot im\right)\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      10. *-commutative 44.8%

          \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(0.25 \cdot im\right)} \]

      11. *-commutative 44.8%

          \[\leadsto 1 + \frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \color{blue}{\left(im \cdot 0.25\right)}} \]

   10. Applied egg-rr 44.8%

       \[\leadsto 1 + \color{blue}{\frac{im \cdot \left(im \cdot 0.25\right) - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* 44.8%

          \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot 0.25} - \left(im \cdot \left(im \cdot 0.25\right)\right) \cdot \left(im \cdot \left(im \cdot 0.25\right)\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       2. swap-sqr 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot 0.25 - \color{blue}{\left(im \cdot im\right) \cdot \left(\left(im \cdot 0.25\right) \cdot \left(im \cdot 0.25\right)\right)}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       3. unpow2 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot 0.25 - \left(im \cdot im\right) \cdot \color{blue}{{\left(im \cdot 0.25\right)}^{2}}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       4. distribute-lft-out-- 44.8%

          \[\leadsto 1 + \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(0.25 - {\left(im \cdot 0.25\right)}^{2}\right)}}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       5. unpow2 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \color{blue}{\left(im \cdot 0.25\right) \cdot \left(im \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       6. swap-sqr 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \color{blue}{\left(im \cdot im\right) \cdot \left(0.25 \cdot 0.25\right)}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       7. metadata-eval 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot \color{blue}{0.0625}\right)}{0.5 \cdot im - im \cdot \left(im \cdot 0.25\right)} \]

       8. *-commutative 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{\color{blue}{im \cdot 0.5} - im \cdot \left(im \cdot 0.25\right)} \]

       9. distribute-lft-out-- 44.8%

          \[\leadsto 1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{\color{blue}{im \cdot \left(0.5 - im \cdot 0.25\right)}} \]

   12. Simplified 44.8%

       \[\leadsto 1 + \color{blue}{\frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}} \]

   if 2.70000000000000006e154 < im

   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 0 100.0%

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

   5. Taylor expanded in re around 0 76.9%

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

   6. Taylor expanded in im around 0 76.9%

      \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
   7. Step-by-step derivation

      1. +-commutative 76.9%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

      2. unpow2 76.9%

         \[\leadsto 1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

      3. associate-*r* 76.9%

         \[\leadsto 1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right) \]

      4. distribute-rgt-out 76.9%

         \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   8. Simplified 76.9%

      \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

   9. Taylor expanded in im around inf 76.9%

      \[\leadsto 1 + \color{blue}{0.25 \cdot {im}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 76.9%

          \[\leadsto 1 + 0.25 \cdot \color{blue}{\left(im \cdot im\right)} \]

       2. *-commutative 76.9%

          \[\leadsto 1 + \color{blue}{\left(im \cdot im\right) \cdot 0.25} \]

       3. associate-*r* 76.9%

          \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]

   11. Simplified 76.9%

       \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{\left(im \cdot im\right) \cdot \left(0.25 - \left(im \cdot im\right) \cdot 0.0625\right)}{im \cdot \left(0.5 - im \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(im \cdot 0.25\right)\\ \end{array} \]

Alternative 9: 47.7% accurate, 44.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 1 + im \cdot \left(im \cdot 0.25\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (+ 1.0 (* im (* im 0.25))))

C

im = abs(im);
double code(double re, double im) {
	return 1.0 + (im * (im * 0.25));
}

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 = 1.0d0 + (im * (im * 0.25d0))
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 1.0 + (im * (im * 0.25));
}

Python

im = abs(im)
def code(re, im):
	return 1.0 + (im * (im * 0.25))

Julia

im = abs(im)
function code(re, im)
	return Float64(1.0 + Float64(im * Float64(im * 0.25)))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 1.0 + (im * (im * 0.25));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(1.0 + N[(im * N[(im * 0.25), $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 77.0%

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

5. Taylor expanded in re around 0 50.3%

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

6. Taylor expanded in im around 0 51.8%

   \[\leadsto \color{blue}{1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]
7. Step-by-step derivation

   1. +-commutative 51.8%

      \[\leadsto 1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)} \]

   2. unpow2 51.8%

      \[\leadsto 1 + \left(0.5 \cdot im + 0.25 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   3. associate-*r* 51.8%

      \[\leadsto 1 + \left(0.5 \cdot im + \color{blue}{\left(0.25 \cdot im\right) \cdot im}\right) \]

   4. distribute-rgt-out 51.8%

      \[\leadsto 1 + \color{blue}{im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

8. Simplified 51.8%

   \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + 0.25 \cdot im\right)} \]

9. Taylor expanded in im around inf 52.1%

   \[\leadsto 1 + \color{blue}{0.25 \cdot {im}^{2}} \]
10. Step-by-step derivation

    1. unpow2 52.1%

       \[\leadsto 1 + 0.25 \cdot \color{blue}{\left(im \cdot im\right)} \]

    2. *-commutative 52.1%

       \[\leadsto 1 + \color{blue}{\left(im \cdot im\right) \cdot 0.25} \]

    3. associate-*r* 52.1%

       \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]

11. Simplified 52.1%

    \[\leadsto 1 + \color{blue}{im \cdot \left(im \cdot 0.25\right)} \]

12. Final simplification 52.1%

    \[\leadsto 1 + im \cdot \left(im \cdot 0.25\right) \]

Alternative 10: 29.4% accurate, 61.6× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 1 + 0.5 \cdot im \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (+ 1.0 (* 0.5 im)))

C

im = abs(im);
double code(double re, double im) {
	return 1.0 + (0.5 * 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 = 1.0d0 + (0.5d0 * im)
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 1.0 + (0.5 * im);
}

Python

im = abs(im)
def code(re, im):
	return 1.0 + (0.5 * im)

Julia

im = abs(im)
function code(re, im)
	return Float64(1.0 + Float64(0.5 * im))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 1.0 + (0.5 * im);
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(1.0 + N[(0.5 * im), $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 77.0%

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

5. Taylor expanded in re around 0 50.3%

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

6. Taylor expanded in im around 0 31.6%

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

7. Final simplification 31.6%

   \[\leadsto 1 + 0.5 \cdot im \]

Alternative 11: 28.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 1 \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 1.0)

C

im = abs(im);
double code(double re, double im) {
	return 1.0;
}

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 = 1.0d0
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 1.0;
}

Python

im = abs(im)
def code(re, im):
	return 1.0

Julia

im = abs(im)
function code(re, im)
	return 1.0
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

NOTE: im should be positive before calling this function
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. 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 77.0%

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

5. Taylor expanded in re around 0 50.3%

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

6. Taylor expanded in im around 0 31.7%

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

7. Final simplification 31.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))