math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 14

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 3.6)
     (* t_0 (fma im im 2.0))
     (if (<= im 5e+102)
       (* 0.5 (- (+ (exp im) 1.0) im))
       (*
        t_0
        (+
         (- 1.0 im)
         (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 3.6) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 5e+102) {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 3.6)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 5e+102)
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.6], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+102], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. +-commutative 87.0%

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

      2. unpow2 87.0%

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

      3. fma-define 87.0%

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

   5. Simplified 87.0%

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

   if 3.60000000000000009 < im < 5e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 71.4%

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

   if 5e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp im) (- 1.0 im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(im) + (1.0 - 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) + (1.0d0 - im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(im) + (1.0 - im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(im) + (1.0 - im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(im) + Float64(1.0 - im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(im) + (1.0 - im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 78.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 78.1%

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

   2. unsub-neg 78.1%

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

5. Simplified 78.1%

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

6. Final simplification 78.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 - im\right)\right) \]
7. Add Preprocessing

Alternative 4: 72.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.9 \lor \neg \left(im \leq 10^{+103}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 6.9) (not (<= im 1e+103)))
   (*
    (* 0.5 (cos re))
    (+
     (- 1.0 im)
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 6.9) || !(im <= 1e+103)) {
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = 0.5 * ((exp(im) + 1.0) - 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 <= 6.9d0) .or. (.not. (im <= 1d+103))) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 6.9) || !(im <= 1e+103)) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 6.9) or not (im <= 1e+103):
		tmp = (0.5 * math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	else:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 6.9) || !(im <= 1e+103))
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	else
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 6.9) || ~((im <= 1e+103)))
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 6.9], N[Not[LessEqual[im, 1e+103]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.9000000000000004 or 1e103 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 76.1%

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

      2. unsub-neg 76.1%

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

   5. Simplified 76.1%

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

   6. Taylor expanded in im around 0 74.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right)\right) \]

   8. Simplified 74.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)}\right) \]

   if 6.9000000000000004 < im < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + e^{im}\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.9 \lor \neg \left(im \leq 10^{+103}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.5 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 2.5) (not (<= im 1.9e+154)))
   (* 0.5 (* (cos re) (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im)))
   (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 2.5) || !(im <= 1.9e+154)) {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	} else {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 2.5d0) .or. (.not. (im <= 1.9d+154))) then
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im))
    else
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 2.5) || !(im <= 1.9e+154)) {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	} else {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 2.5) or not (im <= 1.9e+154):
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im))
	else:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 2.5) || !(im <= 1.9e+154))
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im)));
	else
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 2.5) || ~((im <= 1.9e+154)))
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 2.5], N[Not[LessEqual[im, 1.9e+154]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.5 or 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 75.0%

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

      2. unsub-neg 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0 88.5%

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

   7. Taylor expanded in re around inf 88.5%

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

   if 2.5 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 71.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + e^{im}\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5 \lor \neg \left(im \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 540:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;2 - {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 540.0)
   (cos re)
   (if (<= im 1.25e+73)
     (- 2.0 (pow re 2.0))
     (* (pow im 4.0) 0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 540.0) {
		tmp = cos(re);
	} else if (im <= 1.25e+73) {
		tmp = 2.0 - pow(re, 2.0);
	} else {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 540.0d0) then
        tmp = cos(re)
    else if (im <= 1.25d+73) then
        tmp = 2.0d0 - (re ** 2.0d0)
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 540.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.25e+73) {
		tmp = 2.0 - Math.pow(re, 2.0);
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 540.0:
		tmp = math.cos(re)
	elif im <= 1.25e+73:
		tmp = 2.0 - math.pow(re, 2.0)
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 540.0)
		tmp = cos(re);
	elseif (im <= 1.25e+73)
		tmp = Float64(2.0 - (re ^ 2.0));
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 540.0)
		tmp = cos(re);
	elseif (im <= 1.25e+73)
		tmp = 2.0 - (re ^ 2.0);
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 540.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.25e+73], N[(2.0 - N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 540

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.3%

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

   if 540 < im < 1.24999999999999994e73

   1. Initial program 100.0%

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

   4. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\cos re + \cos re} \]
   5. Step-by-step derivation

      1. count-2 3.1%

         \[\leadsto \color{blue}{2 \cdot \cos re} \]

   6. Simplified 3.1%

      \[\leadsto \color{blue}{2 \cdot \cos re} \]

   7. Taylor expanded in re around 0 29.6%

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

   if 1.24999999999999994e73 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 98.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right) + 2\right)} \]

      2. distribute-lft-in 98.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot 1 + {im}^{2} \cdot \left(0.08333333333333333 \cdot {im}^{2}\right)\right)} + 2\right) \]

      3. *-rgt-identity 98.0%

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

      4. associate-+l+ 98.0%

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

      5. unpow2 98.0%

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

      6. fma-define 98.0%

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

      7. *-commutative 98.0%

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

      8. associate-*l* 98.0%

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

      9. fma-define 98.0%

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

      10. pow-sqr 98.0%

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

      11. metadata-eval 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in im around inf 98.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \cos re\right) \cdot 0.041666666666666664} \]

      2. associate-*r* 98.0%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)} \]

   8. Simplified 98.0%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)} \]

   9. Taylor expanded in re around 0 63.9%

      \[\leadsto {im}^{4} \cdot \color{blue}{0.041666666666666664} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 540:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;2 - {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.8) (cos re) (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.8) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((exp(im) + 1.0) - 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 <= 3.8d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.8) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.8:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.8)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.8)
		tmp = cos(re);
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.8], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 70.3%

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

   if 3.7999999999999998 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 66.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(1 + e^{im}\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{+26}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.7e+26) (cos re) (* (pow im 4.0) 0.041666666666666664)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.7e+26) {
		tmp = cos(re);
	} else {
		tmp = pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.7d+26) then
        tmp = cos(re)
    else
        tmp = (im ** 4.0d0) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.7e+26) {
		tmp = Math.cos(re);
	} else {
		tmp = Math.pow(im, 4.0) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.7e+26:
		tmp = math.cos(re)
	else:
		tmp = math.pow(im, 4.0) * 0.041666666666666664
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.7e+26)
		tmp = cos(re);
	else
		tmp = Float64((im ^ 4.0) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.7e+26)
		tmp = cos(re);
	else
		tmp = (im ^ 4.0) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.7e+26], N[Cos[re], $MachinePrecision], N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.6999999999999998e26

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.0%

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

   if 4.6999999999999998e26 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 75.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {im}^{2}\right) + 2\right)} \]

      2. distribute-lft-in 75.7%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot 1 + {im}^{2} \cdot \left(0.08333333333333333 \cdot {im}^{2}\right)\right)} + 2\right) \]

      3. *-rgt-identity 75.7%

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

      4. associate-+l+ 75.7%

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

      5. unpow2 75.7%

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

      6. fma-define 75.7%

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

      7. *-commutative 75.7%

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

      8. associate-*l* 75.7%

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

      9. fma-define 75.7%

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

      10. pow-sqr 75.7%

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

      11. metadata-eval 75.7%

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

   5. Simplified 75.7%

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

   6. Taylor expanded in im around inf 75.7%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 75.7%

         \[\leadsto \color{blue}{\left({im}^{4} \cdot \cos re\right) \cdot 0.041666666666666664} \]

      2. associate-*r* 75.7%

         \[\leadsto \color{blue}{{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)} \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{{im}^{4} \cdot \left(\cos re \cdot 0.041666666666666664\right)} \]

   9. Taylor expanded in re around 0 49.4%

      \[\leadsto {im}^{4} \cdot \color{blue}{0.041666666666666664} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.5e+54)
   (cos re)
   (*
    0.5
    (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+54) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - 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 <= 8.5d+54) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.5e+54) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.5e+54:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.5e+54)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.5e+54)
		tmp = cos(re);
	else
		tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.5e+54], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.4999999999999995e54

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.5%

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

   if 8.4999999999999995e54 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 66.0%

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

   7. Taylor expanded in im around 0 56.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} - im\right) \]
   8. Step-by-step derivation

      1. *-commutative 56.0%

         \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) - im\right) \]

   9. Simplified 56.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} - im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 44.7% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (- (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))) im)))

C

double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))) - im)
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
}

Python

def code(re, im):
	return 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))) - im);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 78.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 78.1%

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

   2. unsub-neg 78.1%

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

5. Simplified 78.1%

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

6. Taylor expanded in re around 0 46.9%

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

7. Taylor expanded in im around 0 44.2%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} - im\right) \]
8. Step-by-step derivation

   1. *-commutative 44.2%

      \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) - im\right) \]

9. Simplified 44.2%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} - im\right) \]
10. Add Preprocessing

Alternative 11: 47.9% accurate, 23.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im)))

C

double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * 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 * (1.0d0 + (0.5d0 * im)))) - im)
end function

Java

public static double code(double re, double im) {
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
}

Python

def code(re, im):
	return 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im)

Julia

function code(re, im)
	return Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 78.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
4. Step-by-step derivation

   1. neg-mul-1 78.1%

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

   2. unsub-neg 78.1%

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

5. Simplified 78.1%

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

6. Taylor expanded in im around 0 78.1%

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

7. Taylor expanded in re around 0 48.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)} \]
8. Add Preprocessing

Alternative 12: 29.0% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 63.9%

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

4. Taylor expanded in im around 0 48.0%

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

   1. +-commutative 78.3%

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

   2. unpow2 78.3%

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

   3. fma-define 78.3%

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

6. Simplified 48.0%

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

7. Taylor expanded in im around 0 31.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Alternative 13: 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. Add Preprocessing

3. Taylor expanded in re around 0 63.9%

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

5. Applied egg-rr 9.4%

   \[\leadsto 0.5 \cdot \color{blue}{1.5} \]
6. Step-by-step derivation

   1. metadata-eval 9.4%

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

7. Applied egg-rr 9.4%

   \[\leadsto \color{blue}{0.75} \]
8. Add Preprocessing

Alternative 14: 2.4% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.4%

   \[\leadsto \color{blue}{\log \left({1}^{\cos re}\right)} \]
5. Step-by-step derivation

   1. pow-base-1 2.4%

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

   2. metadata-eval 2.4%

      \[\leadsto \color{blue}{0} \]

6. Simplified 2.4%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))