math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 18

Speedup: 1.0×

• 50.1% 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: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.155)
     (*
      t_0
      (+ (+ 1.0 (* im (+ (* 0.5 im) -1.0))) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1e+103)
       (* 0.5 (+ (exp (- im)) (exp 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 <= 0.155) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	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 = 0.5d0 * cos(re)
    if (im <= 0.155d0) then
        tmp = t_0 * ((1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 0.155) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 0.155:
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(-im) + math.exp(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 <= 0.155)
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(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

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 0.155)
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.155], N[(t$95$0 * N[(N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $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 < 0.154999999999999999

   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 88.6%

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

   4. Taylor expanded in im around 0 88.2%

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

   5. Taylor expanded in im around 0 80.9%

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

   if 0.154999999999999999 < 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 re around 0 91.7%

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

   if 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   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 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.155:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% 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 70.5%

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

   1. neg-mul-1 70.5%

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

   2. unsub-neg 70.5%

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

5. Simplified 70.5%

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

6. Final simplification 70.5%

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

Alternative 4: 73.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(im) + 1.0);
}

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + 1.0), $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 70.5%

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

   1. neg-mul-1 70.5%

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

   2. unsub-neg 70.5%

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

5. Simplified 70.5%

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

6. Taylor expanded in im around 0 68.9%

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

7. Final simplification 68.9%

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

Alternative 5: 85.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 0.88)
     (*
      t_0
      (+ (+ 1.0 (* im (+ (* 0.5 im) -1.0))) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1e+103)
       (*
        0.5
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (*
        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 <= 0.88) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	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 = 0.5d0 * cos(re)
    if (im <= 0.88d0) then
        tmp = t_0 * ((1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 0.88) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 0.88:
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	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 <= 0.88)
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	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

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 0.88)
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.88], N[(t$95$0 * N[(N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 0.880000000000000004

   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 88.6%

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

   4. Taylor expanded in im around 0 88.2%

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

   5. Taylor expanded in im around 0 80.9%

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

   if 0.880000000000000004 < 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 re around 0 91.7%

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

   4. Taylor expanded in im around 0 88.7%

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

   if 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   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 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 3.6e-5)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1e+103)
       (*
        0.5
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (*
        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 <= 3.6e-5) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	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 = 0.5d0 * cos(re)
    if (im <= 3.6d-5) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 3.6e-5) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 3.6e-5:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	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 <= 3.6e-5)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	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

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 3.6e-5)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.6e-5], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 3.60000000000000009e-5

   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 60.2%

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

      1. neg-mul-1 60.2%

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

      2. unsub-neg 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in im around 0 81.2%

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

   if 3.60000000000000009e-5 < im < 1e103

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 87.5%

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

   4. Taylor expanded in im around 0 83.0%

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

   if 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + -1 \cdot im\right)} + e^{im}\right) \]
   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 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := im \cdot \left(1 + 0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + t\_1\right)\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* im (+ 1.0 (* 0.5 im)))))
   (if (<= im 0.88)
     (* t_0 (+ (- 1.0 im) (+ 1.0 t_1)))
     (if (<= im 1.85e+154)
       (* 0.5 (- (+ (exp im) 1.0) im))
       (* t_0 (+ t_1 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = im * (1.0 + (0.5 * im));
	double tmp;
	if (im <= 0.88) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1.85e+154) {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	} else {
		tmp = t_0 * (t_1 + 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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = im * (1.0d0 + (0.5d0 * im))
    if (im <= 0.88d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + t_1))
    else if (im <= 1.85d+154) then
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    else
        tmp = t_0 * (t_1 + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = im * (1.0 + (0.5 * im));
	double tmp;
	if (im <= 0.88) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 1.85e+154) {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	} else {
		tmp = t_0 * (t_1 + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = im * (1.0 + (0.5 * im))
	tmp = 0
	if im <= 0.88:
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1))
	elif im <= 1.85e+154:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	else:
		tmp = t_0 * (t_1 + 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(im * Float64(1.0 + Float64(0.5 * im)))
	tmp = 0.0
	if (im <= 0.88)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + t_1)));
	elseif (im <= 1.85e+154)
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	else
		tmp = Float64(t_0 * Float64(t_1 + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = im * (1.0 + (0.5 * im));
	tmp = 0.0;
	if (im <= 0.88)
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	elseif (im <= 1.85e+154)
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	else
		tmp = t_0 * (t_1 + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.88], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.85e+154], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.880000000000000004

   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 59.8%

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

      1. neg-mul-1 59.8%

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

      2. unsub-neg 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in im around 0 80.2%

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

   if 0.880000000000000004 < im < 1.84999999999999997e154

   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%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in re around 0 77.5%

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

   if 1.84999999999999997e154 < 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%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.88)
   (cos re)
   (if (<= im 1.85e+154)
     (* 0.5 (- (+ (exp im) 1.0) im))
     (* (* 0.5 (cos re)) (+ (* im (+ 1.0 (* 0.5 im))) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.88) {
		tmp = cos(re);
	} else if (im <= 1.85e+154) {
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	} else {
		tmp = (0.5 * cos(re)) * ((im * (1.0 + (0.5 * im))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.88d0) then
        tmp = cos(re)
    else if (im <= 1.85d+154) then
        tmp = 0.5d0 * ((exp(im) + 1.0d0) - im)
    else
        tmp = (0.5d0 * cos(re)) * ((im * (1.0d0 + (0.5d0 * im))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.88) {
		tmp = Math.cos(re);
	} else if (im <= 1.85e+154) {
		tmp = 0.5 * ((Math.exp(im) + 1.0) - im);
	} else {
		tmp = (0.5 * Math.cos(re)) * ((im * (1.0 + (0.5 * im))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.88:
		tmp = math.cos(re)
	elif im <= 1.85e+154:
		tmp = 0.5 * ((math.exp(im) + 1.0) - im)
	else:
		tmp = (0.5 * math.cos(re)) * ((im * (1.0 + (0.5 * im))) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.88)
		tmp = cos(re);
	elseif (im <= 1.85e+154)
		tmp = Float64(0.5 * Float64(Float64(exp(im) + 1.0) - im));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.88)
		tmp = cos(re);
	elseif (im <= 1.85e+154)
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	else
		tmp = (0.5 * cos(re)) * ((im * (1.0 + (0.5 * im))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.88], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.85e+154], N[(0.5 * N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.880000000000000004

   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 58.8%

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

   if 0.880000000000000004 < im < 1.84999999999999997e154

   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%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in re around 0 77.5%

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

   if 1.84999999999999997e154 < 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%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.85:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.85) (cos re) (* 0.5 (- (+ (exp im) 1.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.85) {
		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 <= 0.85d0) 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 <= 0.85) {
		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 <= 0.85:
		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 <= 0.85)
		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 <= 0.85)
		tmp = cos(re);
	else
		tmp = 0.5 * ((exp(im) + 1.0) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.85], 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 < 0.849999999999999978

   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 58.8%

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

   if 0.849999999999999978 < 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 98.9%

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

      1. neg-mul-1 98.9%

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

      2. unsub-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in re around 0 84.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.85:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{im} + 1\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.88:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.88) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.88) {
		tmp = cos(re);
	} else {
		tmp = 0.5 + (0.5 * 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 <= 0.88d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.88) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.88:
		tmp = math.cos(re)
	else:
		tmp = 0.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.88)
		tmp = cos(re);
	else
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.88)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.88], N[Cos[re], $MachinePrecision], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.880000000000000004

   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 58.8%

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

   if 0.880000000000000004 < 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 98.9%

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

      1. neg-mul-1 98.9%

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

      2. unsub-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in im around 0 98.9%

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

   7. Taylor expanded in re around 0 84.6%

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

      1. distribute-lft-in 84.6%

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

      2. metadata-eval 84.6%

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

   9. Simplified 84.6%

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

Alternative 11: 62.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{+27}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3e+27)
   (cos re)
   (*
    0.5
    (-
     (+ 1.0 (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
     im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3e+27) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * ((1.0 + (1.0 + (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 <= 3d+27) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * ((1.0d0 + (1.0d0 + (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 <= 3e+27) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * ((1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3e+27:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * ((1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3e+27)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(1.0 + 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 <= 3e+27)
		tmp = cos(re);
	else
		tmp = 0.5 * ((1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3e+27], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[(1.0 + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.99999999999999976e27

   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 56.3%

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

   if 2.99999999999999976e27 < 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 85.2%

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

   7. Taylor expanded in im around 0 63.8%

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

      1. *-commutative 77.0%

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

   9. Simplified 63.8%

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

Alternative 12: 44.2% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(t\_0 + 2\right) - im\right)\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + \left(1 + t\_0\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= re 4.4e+122)
     (* 0.5 (- (+ t_0 2.0) im))
     (if (<= re 4.1e+214)
       (*
        0.5
        (+
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
         3.0))
       (* 0.5 (- (+ 1.0 (+ 1.0 t_0)) im))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (re <= 4.4e+122) {
		tmp = 0.5 * ((t_0 + 2.0) - im);
	} else if (re <= 4.1e+214) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	} else {
		tmp = 0.5 * ((1.0 + (1.0 + t_0)) - im);
	}
	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 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (re <= 4.4d+122) then
        tmp = 0.5d0 * ((t_0 + 2.0d0) - im)
    else if (re <= 4.1d+214) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + 3.0d0)
    else
        tmp = 0.5d0 * ((1.0d0 + (1.0d0 + t_0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (re <= 4.4e+122) {
		tmp = 0.5 * ((t_0 + 2.0) - im);
	} else if (re <= 4.1e+214) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	} else {
		tmp = 0.5 * ((1.0 + (1.0 + t_0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if re <= 4.4e+122:
		tmp = 0.5 * ((t_0 + 2.0) - im)
	elif re <= 4.1e+214:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0)
	else:
		tmp = 0.5 * ((1.0 + (1.0 + t_0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (re <= 4.4e+122)
		tmp = Float64(0.5 * Float64(Float64(t_0 + 2.0) - im));
	elseif (re <= 4.1e+214)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + 3.0));
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(1.0 + t_0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (re <= 4.4e+122)
		tmp = 0.5 * ((t_0 + 2.0) - im);
	elseif (re <= 4.1e+214)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	else
		tmp = 0.5 * ((1.0 + (1.0 + t_0)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 4.4e+122], N[(0.5 * N[(N[(t$95$0 + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.1e+214], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(1.0 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.3999999999999998e122

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in re around 0 53.3%

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

   7. Taylor expanded in im around 0 48.4%

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

      1. *-commutative 48.4%

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

   9. Simplified 48.4%

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

   if 4.3999999999999998e122 < re < 4.1e214

   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 11.4%

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

   4. Taylor expanded in im around 0 3.7%

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

   6. Applied egg-rr 33.7%

      \[\leadsto 0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right) + \color{blue}{3}\right) \]

   if 4.1e214 < re

   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 76.2%

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

      1. neg-mul-1 76.2%

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

      2. unsub-neg 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in re around 0 31.8%

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

   7. Taylor expanded in im around 0 24.2%

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

      1. *-commutative 68.3%

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

   9. Simplified 24.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{+214}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+122} \lor \neg \left(re \leq 4.1 \cdot 10^{+214}\right):\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 4.4e+122) (not (<= re 4.1e+214)))
   (*
    0.5
    (- (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0) im))
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     3.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 4.4e+122) || !(re <= 4.1e+214)) {
		tmp = 0.5 * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im);
	} else {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 4.4d+122) .or. (.not. (re <= 4.1d+214))) then
        tmp = 0.5d0 * (((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0) - im)
    else
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 4.4e+122) || !(re <= 4.1e+214)) {
		tmp = 0.5 * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im);
	} else {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 4.4e+122) or not (re <= 4.1e+214):
		tmp = 0.5 * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im)
	else:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 4.4e+122) || !(re <= 4.1e+214))
		tmp = Float64(0.5 * Float64(Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0) - im));
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 4.4e+122) || ~((re <= 4.1e+214)))
		tmp = 0.5 * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im);
	else
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 4.4e+122], N[Not[LessEqual[re, 4.1e+214]], $MachinePrecision]], N[(0.5 * N[(N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.3999999999999998e122 or 4.1e214 < re

   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 70.5%

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

      1. neg-mul-1 70.5%

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

      2. unsub-neg 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in re around 0 50.8%

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

   7. Taylor expanded in im around 0 45.6%

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

      1. *-commutative 45.6%

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

   9. Simplified 45.6%

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

   if 4.3999999999999998e122 < re < 4.1e214

   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 11.4%

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

   4. Taylor expanded in im around 0 3.7%

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

   6. Applied egg-rr 33.7%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+122} \lor \neg \left(re \leq 4.1 \cdot 10^{+214}\right):\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.7% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(0.5 \cdot im\right)\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.4e+122)
   (* 0.5 (- (+ (* im (+ 1.0 (* 0.5 im))) 2.0) im))
   (if (<= re 1.55e+236)
     (*
      0.5
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       3.0))
     (* 0.5 (- (+ 2.0 (* im (* 0.5 im))) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.4e+122) {
		tmp = 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
	} else if (re <= 1.55e+236) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	} else {
		tmp = 0.5 * ((2.0 + (im * (0.5 * im))) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.4d+122) then
        tmp = 0.5d0 * (((im * (1.0d0 + (0.5d0 * im))) + 2.0d0) - im)
    else if (re <= 1.55d+236) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + 3.0d0)
    else
        tmp = 0.5d0 * ((2.0d0 + (im * (0.5d0 * im))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.4e+122) {
		tmp = 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
	} else if (re <= 1.55e+236) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	} else {
		tmp = 0.5 * ((2.0 + (im * (0.5 * im))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.4e+122:
		tmp = 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im)
	elif re <= 1.55e+236:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0)
	else:
		tmp = 0.5 * ((2.0 + (im * (0.5 * im))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.4e+122)
		tmp = Float64(0.5 * Float64(Float64(Float64(im * Float64(1.0 + Float64(0.5 * im))) + 2.0) - im));
	elseif (re <= 1.55e+236)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + 3.0));
	else
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(0.5 * im))) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.4e+122)
		tmp = 0.5 * (((im * (1.0 + (0.5 * im))) + 2.0) - im);
	elseif (re <= 1.55e+236)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + 3.0);
	else
		tmp = 0.5 * ((2.0 + (im * (0.5 * im))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.4e+122], N[(0.5 * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e+236], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(2.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < 4.3999999999999998e122

   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 69.7%

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

      1. neg-mul-1 69.7%

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

      2. unsub-neg 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in re around 0 53.3%

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

   7. Taylor expanded in im around 0 54.6%

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

   if 4.3999999999999998e122 < re < 1.55e236

   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 20.6%

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

   4. Taylor expanded in im around 0 15.3%

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

   6. Applied egg-rr 30.4%

      \[\leadsto 0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + -0.16666666666666666 \cdot im\right) - 1\right)\right) + \color{blue}{3}\right) \]

   if 1.55e236 < re

   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 74.0%

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

      1. neg-mul-1 74.0%

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

      2. unsub-neg 74.0%

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

   5. Simplified 74.0%

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

   6. Taylor expanded in re around 0 33.8%

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

   7. Taylor expanded in im around 0 34.3%

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

   8. Taylor expanded in im around inf 34.3%

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

      1. *-commutative 34.3%

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

   10. Simplified 34.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right)\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{+236}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + 3\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(0.5 \cdot im\right)\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.4% accurate, 23.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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 70.5%

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

   1. neg-mul-1 70.5%

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

   2. unsub-neg 70.5%

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

5. Simplified 70.5%

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

6. Taylor expanded in re around 0 48.8%

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

7. Taylor expanded in im around 0 49.2%

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

8. Final simplification 49.2%

   \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(1 + 0.5 \cdot im\right) + 2\right) - im\right) \]
9. Add Preprocessing

Alternative 16: 47.1% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(N[(2.0 + N[(im * N[(0.5 * im), $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 70.5%

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

   1. neg-mul-1 70.5%

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

   2. unsub-neg 70.5%

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

5. Simplified 70.5%

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

6. Taylor expanded in re around 0 48.8%

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

7. Taylor expanded in im around 0 49.2%

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

8. Taylor expanded in im around inf 48.8%

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

   1. *-commutative 48.8%

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

10. Simplified 48.8%

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

11. Final simplification 48.8%

    \[\leadsto 0.5 \cdot \left(\left(2 + im \cdot \left(0.5 \cdot im\right)\right) - im\right) \]
12. Add Preprocessing

Alternative 17: 28.3% 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 im around 0 70.5%

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

   1. neg-mul-1 70.5%

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

   2. unsub-neg 70.5%

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

5. Simplified 70.5%

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

6. Taylor expanded in re around 0 48.8%

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

7. Taylor expanded in im around 0 26.0%

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

Alternative 18: 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. Add Preprocessing

3. Taylor expanded in re around 0 70.8%

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

5. Applied egg-rr 3.0%

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

   1. metadata-eval 3.0%

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

7. Applied egg-rr 3.0%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))