math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 11.0s

Alternatives: 20

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.036 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.036) (not (<= im 1.32e+154)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (* (+ (exp (- im)) (exp im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.036) || !(im <= 1.32e+154)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 0.036d0) .or. (.not. (im <= 1.32d+154))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = (exp(-im) + exp(im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.036) || !(im <= 1.32e+154)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.036) or not (im <= 1.32e+154):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.036) || !(im <= 1.32e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.036) || ~((im <= 1.32e+154)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = (exp(-im) + exp(im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.036], N[Not[LessEqual[im, 1.32e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0359999999999999973 or 1.31999999999999998e154 < 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 84.6%

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

   4. Simplified 84.6%

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

   if 0.0359999999999999973 < im < 1.31999999999999998e154

   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 0.0%

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

   4. Simplified 77.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.036 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 3: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.032 \lor \neg \left(im \leq 1.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.032) (not (<= im 1.55e+152)))
   (* (* 0.5 (cos re)) (+ 2.0 (* im im)))
   (* 0.5 (+ (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.032) || !(im <= 1.55e+152)) {
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (exp(-im) + exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 0.032d0) .or. (.not. (im <= 1.55d+152))) then
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * im))
    else
        tmp = 0.5d0 * (exp(-im) + exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 0.032) || !(im <= 1.55e+152)) {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * im));
	} else {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 0.032) or not (im <= 1.55e+152):
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * im))
	else:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.032) || !(im <= 1.55e+152))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 0.032) || ~((im <= 1.55e+152)))
		tmp = (0.5 * cos(re)) * (2.0 + (im * im));
	else
		tmp = 0.5 * (exp(-im) + exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.032], N[Not[LessEqual[im, 1.55e+152]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $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 < 0.032000000000000001 or 1.55e152 < 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 84.3%

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

   4. Simplified 84.3%

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

   if 0.032000000000000001 < im < 1.55e152

   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 71.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.032 \lor \neg \left(im \leq 1.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 4: 81.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{if}\;im \leq 31000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 10^{+77}:\\ \;\;\;\;t_0 \cdot \left(0.5 + {re}^{-2}\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.5 + re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))) (t_1 (* (* 0.5 (cos re)) t_0)))
   (if (<= im 31000.0)
     t_1
     (if (<= im 1e+77)
       (* t_0 (+ 0.5 (pow re -2.0)))
       (if (<= im 1.55e+152)
         (/ (+ 0.5 re) (/ (- 2.0 (* im im)) (- 4.0 (pow im 4.0))))
         t_1)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * cos(re)) * t_0;
	double tmp;
	if (im <= 31000.0) {
		tmp = t_1;
	} else if (im <= 1e+77) {
		tmp = t_0 * (0.5 + pow(re, -2.0));
	} else if (im <= 1.55e+152) {
		tmp = (0.5 + re) / ((2.0 - (im * im)) / (4.0 - pow(im, 4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = (0.5d0 * cos(re)) * t_0
    if (im <= 31000.0d0) then
        tmp = t_1
    else if (im <= 1d+77) then
        tmp = t_0 * (0.5d0 + (re ** (-2.0d0)))
    else if (im <= 1.55d+152) then
        tmp = (0.5d0 + re) / ((2.0d0 - (im * im)) / (4.0d0 - (im ** 4.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = (0.5 * Math.cos(re)) * t_0;
	double tmp;
	if (im <= 31000.0) {
		tmp = t_1;
	} else if (im <= 1e+77) {
		tmp = t_0 * (0.5 + Math.pow(re, -2.0));
	} else if (im <= 1.55e+152) {
		tmp = (0.5 + re) / ((2.0 - (im * im)) / (4.0 - Math.pow(im, 4.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = (0.5 * math.cos(re)) * t_0
	tmp = 0
	if im <= 31000.0:
		tmp = t_1
	elif im <= 1e+77:
		tmp = t_0 * (0.5 + math.pow(re, -2.0))
	elif im <= 1.55e+152:
		tmp = (0.5 + re) / ((2.0 - (im * im)) / (4.0 - math.pow(im, 4.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(Float64(0.5 * cos(re)) * t_0)
	tmp = 0.0
	if (im <= 31000.0)
		tmp = t_1;
	elseif (im <= 1e+77)
		tmp = Float64(t_0 * Float64(0.5 + (re ^ -2.0)));
	elseif (im <= 1.55e+152)
		tmp = Float64(Float64(0.5 + re) / Float64(Float64(2.0 - Float64(im * im)) / Float64(4.0 - (im ^ 4.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = (0.5 * cos(re)) * t_0;
	tmp = 0.0;
	if (im <= 31000.0)
		tmp = t_1;
	elseif (im <= 1e+77)
		tmp = t_0 * (0.5 + (re ^ -2.0));
	elseif (im <= 1.55e+152)
		tmp = (0.5 + re) / ((2.0 - (im * im)) / (4.0 - (im ^ 4.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[im, 31000.0], t$95$1, If[LessEqual[im, 1e+77], N[(t$95$0 * N[(0.5 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.55e+152], N[(N[(0.5 + re), $MachinePrecision] / N[(N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 31000 or 1.55e152 < 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 84.3%

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

   4. Simplified 84.3%

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

   if 31000 < im < 9.99999999999999983e76

   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 0.0%

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

   4. Simplified 91.7%

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

   5. Taylor expanded in im around 0 43.6%

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

   7. Simplified 43.6%

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

   9. Applied egg-rr 42.9%

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

   if 9.99999999999999983e76 < im < 1.55e152

   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 0.0%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in im around 0 24.8%

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

   7. Simplified 24.8%

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

   9. Applied egg-rr 25.4%

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

       1. *-commutative 25.4%

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

       2. flip-+ 78.3%

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

       3. associate-*r/ 78.3%

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

       4. add-sqr-sqrt 43.5%

          \[\leadsto \frac{\left(0.5 + \left|\color{blue}{\sqrt{re} \cdot \sqrt{re}}\right|\right) \cdot \left(2 \cdot 2 - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}{2 - im \cdot im} \]

       5. fabs-sqr 43.5%

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

       6. add-sqr-sqrt 69.6%

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

       7. metadata-eval 69.6%

          \[\leadsto \frac{\left(0.5 + re\right) \cdot \left(\color{blue}{4} - \left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}{2 - im \cdot im} \]

       8. pow2 69.6%

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

       9. pow2 69.6%

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

       10. pow-prod-up 69.6%

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

       11. metadata-eval 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. associate-/l* 69.6%

          \[\leadsto \color{blue}{\frac{0.5 + re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}} \]

   13. Simplified 69.6%

       \[\leadsto \color{blue}{\frac{0.5 + re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 31000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+77}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + {re}^{-2}\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.5 + re}{\frac{2 - im \cdot im}{4 - {im}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \end{array} \]

Alternative 5: 80.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 31000 \lor \neg \left(im \leq 1.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + {re}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))))
   (if (or (<= im 31000.0) (not (<= im 1.55e+152)))
     (* (* 0.5 (cos re)) t_0)
     (* t_0 (+ 0.5 (pow re -2.0))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if ((im <= 31000.0) || !(im <= 1.55e+152)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = t_0 * (0.5 + pow(re, -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    if ((im <= 31000.0d0) .or. (.not. (im <= 1.55d+152))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = t_0 * (0.5d0 + (re ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if ((im <= 31000.0) || !(im <= 1.55e+152)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = t_0 * (0.5 + Math.pow(re, -2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	tmp = 0
	if (im <= 31000.0) or not (im <= 1.55e+152):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = t_0 * (0.5 + math.pow(re, -2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if ((im <= 31000.0) || !(im <= 1.55e+152))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(t_0 * Float64(0.5 + (re ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	tmp = 0.0;
	if ((im <= 31000.0) || ~((im <= 1.55e+152)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = t_0 * (0.5 + (re ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[im, 31000.0], N[Not[LessEqual[im, 1.55e+152]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 31000 or 1.55e152 < 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 84.3%

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

   4. Simplified 84.3%

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

   if 31000 < im < 1.55e152

   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 0.0%

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

   4. Simplified 77.1%

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

   5. Taylor expanded in im around 0 31.2%

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

   7. Simplified 31.2%

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

   9. Applied egg-rr 45.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 31000 \lor \neg \left(im \leq 1.55 \cdot 10^{+152}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + {re}^{-2}\right)\\ \end{array} \]

Alternative 6: 77.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 610 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im))))
   (if (or (<= im 610.0) (not (<= im 1.32e+154)))
     (* (* 0.5 (cos re)) t_0)
     (* t_0 (+ 0.5 (* -0.25 (* re re)))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if ((im <= 610.0) || !(im <= 1.32e+154)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    if ((im <= 610.0d0) .or. (.not. (im <= 1.32d+154))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = t_0 * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double tmp;
	if ((im <= 610.0) || !(im <= 1.32e+154)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	tmp = 0
	if (im <= 610.0) or not (im <= 1.32e+154):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = t_0 * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if ((im <= 610.0) || !(im <= 1.32e+154))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(t_0 * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	tmp = 0.0;
	if ((im <= 610.0) || ~((im <= 1.32e+154)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = t_0 * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[im, 610.0], N[Not[LessEqual[im, 1.32e+154]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 610 or 1.31999999999999998e154 < 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 84.6%

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

   4. Simplified 84.6%

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

   if 610 < im < 1.31999999999999998e154

   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 0.0%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in im around 0 33.1%

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

   7. Simplified 33.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 610 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7: 62.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.07:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.07) (cos re) (* (+ 2.0 (* im im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.07) {
		tmp = cos(re);
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.07d0) then
        tmp = cos(re)
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.07) {
		tmp = Math.cos(re);
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.07:
		tmp = math.cos(re)
	else:
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.07)
		tmp = cos(re);
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.07)
		tmp = cos(re);
	else
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.07], N[Cos[re], $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.070000000000000007

   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 71.0%

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

   if 0.070000000000000007 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in im around 0 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.07:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 8: 39.9% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2e-5) 1.0 (* (+ 2.0 (* im im)) (+ 0.5 (* -0.25 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d-5) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 + ((-0.25d0) * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2e-5:
		tmp = 1.0
	else:
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = (2.0 + (im * im)) * (0.5 + (-0.25 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2e-5], 1.0, N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000027e-5

   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 56.6%

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

   3. Taylor expanded in im around 0 36.9%

      \[\leadsto 0.5 \cdot \color{blue}{2} \]

   if 6.20000000000000027e-5 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in im around 0 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 9: 48.4% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 112:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+104}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 512\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 112.0)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 8.8e+104)
     (* (+ 0.5 (* -0.25 (* re re))) 512.0)
     (* (* im im) (+ 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 112.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 8.8e+104) {
		tmp = (0.5 + (-0.25 * (re * re))) * 512.0;
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 112.0d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 8.8d+104) then
        tmp = (0.5d0 + ((-0.25d0) * (re * re))) * 512.0d0
    else
        tmp = (im * im) * (0.5d0 + re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 112.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 8.8e+104) {
		tmp = (0.5 + (-0.25 * (re * re))) * 512.0;
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 112.0:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 8.8e+104:
		tmp = (0.5 + (-0.25 * (re * re))) * 512.0
	else:
		tmp = (im * im) * (0.5 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 112.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 8.8e+104)
		tmp = Float64(Float64(0.5 + Float64(-0.25 * Float64(re * re))) * 512.0);
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 112.0)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 8.8e+104)
		tmp = (0.5 + (-0.25 * (re * re))) * 512.0;
	else
		tmp = (im * im) * (0.5 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 112.0], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.8e+104], N[(N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 512.0), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 112

   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 56.6%

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

   3. Taylor expanded in im around 0 44.4%

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

   5. Simplified 44.4%

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

   if 112 < im < 8.80000000000000002e104

   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 0.0%

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

   4. Simplified 90.5%

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

   6. Applied egg-rr 39.8%

      \[\leadsto \color{blue}{512} \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \]

   if 8.80000000000000002e104 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

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

   4. Simplified 75.6%

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

   5. Taylor expanded in im around 0 63.2%

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

   7. Simplified 63.2%

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

   9. Applied egg-rr 61.2%

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

   10. Taylor expanded in im around inf 61.2%

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

       1. unpow2 61.2%

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

       2. +-commutative 61.2%

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

       3. distribute-lft-in 61.2%

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

       4. fma-def 61.2%

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

       5. rem-square-sqrt 27.6%

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

       6. fabs-sqr 27.6%

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

       7. rem-square-sqrt 32.3%

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

       8. fma-def 32.3%

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

       9. distribute-lft-in 65.6%

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

       10. +-commutative 65.6%

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

       11. *-commutative 65.6%

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

   12. Simplified 65.6%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 112:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+104}:\\ \;\;\;\;\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right) \cdot 512\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \]

Alternative 10: 48.4% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 155:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+105}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 155.0)
   (* 0.5 (+ 2.0 (* im im)))
   (if (<= im 1.45e+105) (+ 1.0 (* (* re re) -0.5)) (* (* im im) (+ 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 155.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 1.45e+105) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 155.0d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else if (im <= 1.45d+105) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = (im * im) * (0.5d0 + re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 155.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else if (im <= 1.45e+105) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 155.0:
		tmp = 0.5 * (2.0 + (im * im))
	elif im <= 1.45e+105:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = (im * im) * (0.5 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 155.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.45e+105)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 155.0)
		tmp = 0.5 * (2.0 + (im * im));
	elseif (im <= 1.45e+105)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = (im * im) * (0.5 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 155.0], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.45e+105], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 155

   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 56.6%

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

   3. Taylor expanded in im around 0 44.4%

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

   5. Simplified 44.4%

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

   if 155 < im < 1.45000000000000005e105

   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 0.0%

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

   4. Simplified 90.5%

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

   5. Taylor expanded in im around 0 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in im around 0 39.7%

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

      1. distribute-rgt-in 39.7%

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

      2. metadata-eval 39.7%

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

      3. unpow2 39.7%

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

      4. *-commutative 39.7%

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

      5. associate-*l* 39.7%

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

      6. metadata-eval 39.7%

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

   10. Simplified 39.7%

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

   if 1.45000000000000005e105 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

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

   4. Simplified 75.6%

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

   5. Taylor expanded in im around 0 63.2%

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

   7. Simplified 63.2%

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

   9. Applied egg-rr 61.2%

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

   10. Taylor expanded in im around inf 61.2%

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

       1. unpow2 61.2%

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

       2. +-commutative 61.2%

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

       3. distribute-lft-in 61.2%

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

       4. fma-def 61.2%

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

       5. rem-square-sqrt 27.6%

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

       6. fabs-sqr 27.6%

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

       7. rem-square-sqrt 32.3%

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

       8. fma-def 32.3%

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

       9. distribute-lft-in 65.6%

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

       10. +-commutative 65.6%

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

       11. *-commutative 65.6%

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

   12. Simplified 65.6%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 155:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{+105}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \]

Alternative 11: 48.1% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 78000000000000:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 78000000000000.0)
   (* 0.5 (+ 2.0 (* im im)))
   (* (* im im) (+ 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 78000000000000.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 78000000000000.0d0) then
        tmp = 0.5d0 * (2.0d0 + (im * im))
    else
        tmp = (im * im) * (0.5d0 + re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 78000000000000.0) {
		tmp = 0.5 * (2.0 + (im * im));
	} else {
		tmp = (im * im) * (0.5 + re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 78000000000000.0:
		tmp = 0.5 * (2.0 + (im * im))
	else:
		tmp = (im * im) * (0.5 + re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 78000000000000.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * im)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 + re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 78000000000000.0)
		tmp = 0.5 * (2.0 + (im * im));
	else
		tmp = (im * im) * (0.5 + re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 78000000000000.0], N[(0.5 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 + re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.8e13

   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 56.3%

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

   3. Taylor expanded in im around 0 44.2%

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

   5. Simplified 44.2%

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

   if 7.8e13 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

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

   4. Simplified 80.0%

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

   5. Taylor expanded in im around 0 55.3%

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

   7. Simplified 55.3%

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

   9. Applied egg-rr 44.7%

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

   10. Taylor expanded in im around inf 44.7%

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

       1. unpow2 44.7%

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

       2. +-commutative 44.7%

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

       3. distribute-lft-in 44.7%

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

       4. fma-def 44.7%

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

       5. rem-square-sqrt 20.9%

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

       6. fabs-sqr 20.9%

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

       7. rem-square-sqrt 31.1%

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

       8. fma-def 31.1%

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

       9. distribute-lft-in 54.1%

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

       10. +-commutative 54.1%

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

       11. *-commutative 54.1%

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

   12. Simplified 54.1%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 78000000000000:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + re\right)\\ \end{array} \]

Alternative 12: 46.5% accurate, 44.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (+ 2.0 (* im im))))

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 re around 0 60.3%

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

3. Taylor expanded in im around 0 42.1%

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

5. Simplified 42.1%

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

6. Final simplification 42.1%

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

Alternative 13: 3.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.0d0
end function

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 60.3%

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

4. Applied egg-rr 4.1%

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

5. Final simplification 4.1%

   \[\leadsto -1 \]

Alternative 14: 6.9% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.0078125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0078125)

C

double code(double re, double im) {
	return 0.0078125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0078125d0
end function

Java

public static double code(double re, double im) {
	return 0.0078125;
}

Python

def code(re, im):
	return 0.0078125

Julia

function code(re, im)
	return 0.0078125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0078125;
end

Wolfram

code[re_, im_] := 0.0078125

Derivation

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 60.3%

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

4. Applied egg-rr 7.0%

   \[\leadsto 0.5 \cdot \color{blue}{0.015625} \]

5. Final simplification 7.0%

   \[\leadsto 0.0078125 \]

Alternative 15: 7.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

Derivation

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 60.3%

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

4. Applied egg-rr 7.9%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]

5. Final simplification 7.9%

   \[\leadsto 0.125 \]

Alternative 16: 8.1% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.25)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

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

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 60.3%

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

4. Applied egg-rr 8.2%

   \[\leadsto 0.5 \cdot \color{blue}{0.5} \]

5. Final simplification 8.2%

   \[\leadsto 0.25 \]

Alternative 17: 8.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.375 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.375)

C

double code(double re, double im) {
	return 0.375;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.375d0
end function

Java

public static double code(double re, double im) {
	return 0.375;
}

Python

def code(re, im):
	return 0.375

Julia

function code(re, im)
	return 0.375
end

MATLAB

function tmp = code(re, im)
	tmp = 0.375;
end

Wolfram

code[re_, im_] := 0.375

Derivation

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 60.3%

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

4. Applied egg-rr 8.5%

   \[\leadsto 0.5 \cdot \color{blue}{0.75} \]

5. Final simplification 8.5%

   \[\leadsto 0.375 \]

Alternative 18: 8.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0
end function

Java

public static double code(double re, double im) {
	return 0.5;
}

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

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 60.3%

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

4. Applied egg-rr 8.8%

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

5. Final simplification 8.8%

   \[\leadsto 0.5 \]

Alternative 19: 9.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.75)

C

double code(double re, double im) {
	return 0.75;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.75d0
end function

Java

public static double code(double re, double im) {
	return 0.75;
}

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

function tmp = code(re, im)
	tmp = 0.75;
end

Wolfram

code[re_, im_] := 0.75

Derivation

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 60.3%

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

4. Applied egg-rr 9.3%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]

5. Final simplification 9.3%

   \[\leadsto 0.75 \]

Alternative 20: 28.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0 60.3%

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

3. Taylor expanded in im around 0 28.0%

   \[\leadsto 0.5 \cdot \color{blue}{2} \]

4. Final simplification 28.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023283 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))