math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

Alternatives: 19

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: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;t\_1 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (cos re))))
   (if (<= im 0.035)
     (*
      t_1
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 2.5e+101)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_1 (+ 2.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.035) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_1 * (2.0 + t_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 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * cos(re)
    if (im <= 0.035d0) then
        tmp = t_1 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = t_1 * (2.0d0 + t_0)
    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 t_1 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 0.035) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = t_1 * (2.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 0.035:
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = t_1 * (2.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.035)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + t_0)));
	elseif (im <= 2.5e+101)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_1 * Float64(2.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 0.035)
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = t_1 * (2.0 + t_0);
	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]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.035], N[(t$95$1 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+101], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.035000000000000003

   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 87.5%

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

   4. Taylor expanded in im around 0 68.9%

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

      1. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   if 0.035000000000000003 < im < 2.49999999999999994e101

   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 66.6%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.035:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% 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 76.5%

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

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Final simplification 76.5%

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

Alternative 4: 74.2% 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 76.5%

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

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in im around 0 75.7%

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

7. Final simplification 75.7%

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

Alternative 5: 72.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ t_2 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (cos re)))
        (t_2
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 1.85)
     (* t_1 (+ t_2 (+ 1.0 t_0)))
     (if (<= im 2.5e+101) (* 0.5 (+ (exp im) t_2)) (* t_1 (+ 2.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * cos(re);
	double t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(im) + t_2);
	} else {
		tmp = t_1 * (2.0 + t_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) :: t_2
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * cos(re)
    t_2 = 1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (im <= 1.85d0) then
        tmp = t_1 * (t_2 + (1.0d0 + t_0))
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(im) + t_2)
    else
        tmp = t_1 * (2.0d0 + t_0)
    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 t_1 = 0.5 * Math.cos(re);
	double t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(im) + t_2);
	} else {
		tmp = t_1 * (2.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_1 = 0.5 * math.cos(re)
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))
	tmp = 0
	if im <= 1.85:
		tmp = t_1 * (t_2 + (1.0 + t_0))
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(im) + t_2)
	else:
		tmp = t_1 * (2.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_1 * Float64(t_2 + Float64(1.0 + t_0)));
	elseif (im <= 2.5e+101)
		tmp = Float64(0.5 * Float64(exp(im) + t_2));
	else
		tmp = Float64(t_1 * Float64(2.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_1 = 0.5 * cos(re);
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_1 * (t_2 + (1.0 + t_0));
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(im) + t_2);
	else
		tmp = t_1 * (2.0 + t_0);
	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]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], N[(t$95$1 * N[(t$95$2 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+101], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.8500000000000001

   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 87.5%

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

   4. Taylor expanded in im around 0 68.9%

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

      1. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   if 1.8500000000000001 < im < 2.49999999999999994e101

   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 66.6%

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

   4. Taylor expanded in im around 0 62.4%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 1.85)
     (* t_0 (+ t_1 (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 2.5e+101)
       (* 0.5 (+ (exp im) t_1))
       (*
        t_0
        (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(im) + t_1);
	} else {
		tmp = t_0 * (2.0 + (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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = 1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (im <= 1.85d0) then
        tmp = t_0 * (t_1 + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(im) + t_1)
    else
        tmp = t_0 * (2.0d0 + (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 t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(im) + t_1);
	} else {
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))
	tmp = 0
	if im <= 1.85:
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(im) + t_1)
	else:
		tmp = t_0 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_0 * Float64(t_1 + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 2.5e+101)
		tmp = Float64(0.5 * Float64(exp(im) + t_1));
	else
		tmp = Float64(t_0 * Float64(2.0 + 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);
	t_1 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_0 * (t_1 + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(im) + t_1);
	else
		tmp = t_0 * (2.0 + (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]}, Block[{t$95$1 = N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], N[(t$95$0 * N[(t$95$1 + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+101], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.8500000000000001

   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 87.5%

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

   4. Taylor expanded in im around 0 87.5%

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

   if 1.8500000000000001 < im < 2.49999999999999994e101

   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 66.6%

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

   4. Taylor expanded in im around 0 62.4%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + t\_1\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 1.85)
     (* t_0 (+ (- 1.0 im) (+ 1.0 t_1)))
     (if (<= im 2.5e+101)
       (*
        0.5
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (* t_0 (+ 2.0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 1.85d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + t_1))
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (2.0d0 + t_1)
    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 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 1.85) {
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 1.85:
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1))
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (2.0 + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + t_1)));
	elseif (im <= 2.5e+101)
		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(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 1.85)
		tmp = t_0 * ((1.0 - im) + (1.0 + t_1));
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (2.0 + t_1);
	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[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.85], N[(t$95$0 * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e+101], 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[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.8500000000000001

   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 70.1%

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

      1. neg-mul-1 70.1%

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

      2. unsub-neg 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0 68.5%

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

      1. *-commutative 68.9%

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

   8. Simplified 68.5%

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

   if 1.8500000000000001 < im < 2.49999999999999994e101

   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 66.6%

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

   4. Taylor expanded in im around 0 62.4%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 1.7)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 2.5e+101)
       (*
        0.5
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (*
        t_0
        (+ 2.0 (* 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 <= 1.7) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + (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 <= 1.7d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (2.0d0 + (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 <= 1.7) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (2.0 + (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 <= 1.7:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (2.0 + (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 <= 1.7)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 2.5e+101)
		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(2.0 + 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 <= 1.7)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (2.0 + (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, 1.7], 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, 2.5e+101], 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[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.69999999999999996

   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 70.1%

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

      1. neg-mul-1 70.1%

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

      2. unsub-neg 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in im around 0 83.5%

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

   if 1.69999999999999996 < im < 2.49999999999999994e101

   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 66.6%

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

   4. Taylor expanded in im around 0 62.4%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0042:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0042)
   (cos re)
   (if (<= im 2.5e+101)
     (*
      0.5
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      (* 0.5 (cos re))
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0042) {
		tmp = cos(re);
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * cos(re)) * (2.0 + (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) :: tmp
    if (im <= 0.0042d0) then
        tmp = cos(re)
    else if (im <= 2.5d+101) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (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 tmp;
	if (im <= 0.0042) {
		tmp = Math.cos(re);
	} else if (im <= 2.5e+101) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0042:
		tmp = math.cos(re)
	elif im <= 2.5e+101:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0042)
		tmp = cos(re);
	elseif (im <= 2.5e+101)
		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(Float64(0.5 * cos(re)) * Float64(2.0 + 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)
	tmp = 0.0;
	if (im <= 0.0042)
		tmp = cos(re);
	elseif (im <= 2.5e+101)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = (0.5 * cos(re)) * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0042], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.5e+101], 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[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00419999999999999974

   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 69.4%

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

   if 0.00419999999999999974 < im < 2.49999999999999994e101

   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 66.6%

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

   4. Taylor expanded in im around 0 62.4%

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

   if 2.49999999999999994e101 < 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%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0042:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.042)
   (cos re)
   (if (<= im 1.86e+154)
     (* 0.5 (+ (exp im) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))
     (* (* 0.5 (cos re)) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.042) {
		tmp = cos(re);
	} else if (im <= 1.86e+154) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = (0.5 * cos(re)) * (2.0 + (im * (1.0 + (0.5 * 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 <= 0.042d0) then
        tmp = cos(re)
    else if (im <= 1.86d+154) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    else
        tmp = (0.5d0 * cos(re)) * (2.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.042) {
		tmp = Math.cos(re);
	} else if (im <= 1.86e+154) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else {
		tmp = (0.5 * Math.cos(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.042:
		tmp = math.cos(re)
	elif im <= 1.86e+154:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))))
	else:
		tmp = (0.5 * math.cos(re)) * (2.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.042)
		tmp = cos(re);
	elseif (im <= 1.86e+154)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.042)
		tmp = cos(re);
	elseif (im <= 1.86e+154)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	else
		tmp = (0.5 * cos(re)) * (2.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.042], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.86e+154], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0420000000000000026

   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 69.4%

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

   if 0.0420000000000000026 < im < 1.86000000000000014e154

   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 62.0%

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

   4. Taylor expanded in im around 0 59.4%

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

   if 1.86000000000000014e154 < 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%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.042:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.86 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.92:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.92)
   (cos re)
   (* 0.5 (+ (exp im) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.92) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.92d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.92) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.92:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.92)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.92)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((0.5 * im) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.92], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.92000000000000004

   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 69.4%

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

   if 0.92000000000000004 < 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 re around 0 67.2%

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

   4. Taylor expanded in im around 0 65.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.92:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		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 <= 1.85d0) 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 <= 1.85) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		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 <= 1.85)
		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 <= 1.85)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85], 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 < 1.8500000000000001

   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 69.4%

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

   if 1.8500000000000001 < 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 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in im around 0 98.7%

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

   7. Taylor expanded in re around 0 65.9%

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

      1. distribute-lft-in 65.9%

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

      2. metadata-eval 65.9%

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

   9. Simplified 65.9%

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

Alternative 13: 63.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+106}:\\ \;\;\;\;2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (cos re)
   (if (<= im 1.75e+106)
     (- 2.0 (* re re))
     (*
      0.5
      (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = cos(re);
	} else if (im <= 1.75e+106) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 0.5 * (2.0 + (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) :: tmp
    if (im <= 1.85d0) then
        tmp = cos(re)
    else if (im <= 1.75d+106) then
        tmp = 2.0d0 - (re * re)
    else
        tmp = 0.5d0 * (2.0d0 + (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 tmp;
	if (im <= 1.85) {
		tmp = Math.cos(re);
	} else if (im <= 1.75e+106) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		tmp = math.cos(re)
	elif im <= 1.75e+106:
		tmp = 2.0 - (re * re)
	else:
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85)
		tmp = cos(re);
	elseif (im <= 1.75e+106)
		tmp = Float64(2.0 - Float64(re * re));
	else
		tmp = Float64(0.5 * Float64(2.0 + 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)
	tmp = 0.0;
	if (im <= 1.85)
		tmp = cos(re);
	elseif (im <= 1.75e+106)
		tmp = 2.0 - (re * re);
	else
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.75e+106], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.8500000000000001

   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 69.4%

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

   if 1.8500000000000001 < im < 1.7499999999999999e106

   1. Initial program 99.9%

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

   4. Applied egg-rr 4.1%

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

      1. count-2 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in re around 0 28.4%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 28.4%

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

      2. unsub-neg 28.4%

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

   9. Simplified 28.4%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 28.4%

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

   11. Applied egg-rr 28.4%

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

   if 1.7499999999999999e106 < 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%

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

   8. Taylor expanded in re around 0 71.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+106}:\\ \;\;\;\;2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;re \leq 1.26 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(t\_0 - im\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+226} \lor \neg \left(re \leq 6.6 \cdot 10^{+268}\right):\\ \;\;\;\;2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (if (<= re 1.26e+158)
     (* 0.5 (- t_0 im))
     (if (or (<= re 2.5e+226) (not (<= re 6.6e+268)))
       (- 2.0 (* re re))
       (* 0.5 t_0)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))));
	double tmp;
	if (re <= 1.26e+158) {
		tmp = 0.5 * (t_0 - im);
	} else if ((re <= 2.5e+226) || !(re <= 6.6e+268)) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))
    if (re <= 1.26d+158) then
        tmp = 0.5d0 * (t_0 - im)
    else if ((re <= 2.5d+226) .or. (.not. (re <= 6.6d+268))) then
        tmp = 2.0d0 - (re * re)
    else
        tmp = 0.5d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))));
	double tmp;
	if (re <= 1.26e+158) {
		tmp = 0.5 * (t_0 - im);
	} else if ((re <= 2.5e+226) || !(re <= 6.6e+268)) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))
	tmp = 0
	if re <= 1.26e+158:
		tmp = 0.5 * (t_0 - im)
	elif (re <= 2.5e+226) or not (re <= 6.6e+268):
		tmp = 2.0 - (re * re)
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))
	tmp = 0.0
	if (re <= 1.26e+158)
		tmp = Float64(0.5 * Float64(t_0 - im));
	elseif ((re <= 2.5e+226) || !(re <= 6.6e+268))
		tmp = Float64(2.0 - Float64(re * re));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))));
	tmp = 0.0;
	if (re <= 1.26e+158)
		tmp = 0.5 * (t_0 - im);
	elseif ((re <= 2.5e+226) || ~((re <= 6.6e+268)))
		tmp = 2.0 - (re * re);
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, 1.26e+158], N[(0.5 * N[(t$95$0 - im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 2.5e+226], N[Not[LessEqual[re, 6.6e+268]], $MachinePrecision]], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 1.2599999999999999e158

   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 77.8%

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

      1. neg-mul-1 77.8%

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

      2. unsub-neg 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in re around 0 53.7%

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

   7. Taylor expanded in im around 0 50.3%

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

   if 1.2599999999999999e158 < re < 2.5000000000000002e226 or 6.6000000000000002e268 < re

   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 10.7%

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

      1. count-2 10.7%

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

   6. Simplified 10.7%

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

   7. Taylor expanded in re around 0 43.0%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.0%

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

      2. unsub-neg 43.0%

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

   9. Simplified 43.0%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 43.0%

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

   11. Applied egg-rr 43.0%

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

   if 2.5000000000000002e226 < re < 6.6000000000000002e268

   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 71.6%

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

      1. neg-mul-1 71.6%

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

      2. unsub-neg 71.6%

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

   5. Simplified 71.6%

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

   6. Taylor expanded in im around 0 70.9%

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

   7. Taylor expanded in im around 0 70.0%

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

   8. Taylor expanded in re around 0 17.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) - im\right)\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+226} \lor \neg \left(re \leq 6.6 \cdot 10^{+268}\right):\\ \;\;\;\;2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{+158} \lor \neg \left(re \leq 2.5 \cdot 10^{+226}\right) \land re \leq 6.6 \cdot 10^{+268}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 1.26e+158) (and (not (<= re 2.5e+226)) (<= re 6.6e+268)))
   (* 0.5 (+ 2.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 1.26e+158) || (!(re <= 2.5e+226) && (re <= 6.6e+268))) {
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= 1.26d+158) .or. (.not. (re <= 2.5d+226)) .and. (re <= 6.6d+268)) then
        tmp = 0.5d0 * (2.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    else
        tmp = 2.0d0 - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 1.26e+158) || (!(re <= 2.5e+226) && (re <= 6.6e+268))) {
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 1.26e+158) or (not (re <= 2.5e+226) and (re <= 6.6e+268)):
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 1.26e+158) || (!(re <= 2.5e+226) && (re <= 6.6e+268)))
		tmp = Float64(0.5 * Float64(2.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 1.26e+158) || (~((re <= 2.5e+226)) && (re <= 6.6e+268)))
		tmp = 0.5 * (2.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 1.26e+158], And[N[Not[LessEqual[re, 2.5e+226]], $MachinePrecision], LessEqual[re, 6.6e+268]]], N[(0.5 * N[(2.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.2599999999999999e158 or 2.5000000000000002e226 < re < 6.6000000000000002e268

   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 77.5%

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

      1. neg-mul-1 77.5%

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

      2. unsub-neg 77.5%

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

   5. Simplified 77.5%

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

   6. Taylor expanded in im around 0 76.7%

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

   7. Taylor expanded in im around 0 69.7%

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

   8. Taylor expanded in re around 0 48.6%

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

   if 1.2599999999999999e158 < re < 2.5000000000000002e226 or 6.6000000000000002e268 < re

   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 10.7%

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

      1. count-2 10.7%

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

   6. Simplified 10.7%

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

   7. Taylor expanded in re around 0 43.0%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.0%

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

      2. unsub-neg 43.0%

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

   9. Simplified 43.0%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 43.0%

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

   11. Applied egg-rr 43.0%

       \[\leadsto 2 - \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{+158} \lor \neg \left(re \leq 2.5 \cdot 10^{+226}\right) \land re \leq 6.6 \cdot 10^{+268}:\\ \;\;\;\;0.5 \cdot \left(2 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.9% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.26e+158)
   (* 0.5 (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im))
   (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.26e+158) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.26d+158) then
        tmp = 0.5d0 * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im)
    else
        tmp = 2.0d0 - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.26e+158) {
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.26e+158:
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im)
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.26e+158)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im));
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.26e+158)
		tmp = 0.5 * ((2.0 + (im * (1.0 + (0.5 * im)))) - im);
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.26e+158], N[(0.5 * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.2599999999999999e158

   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 77.8%

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

      1. neg-mul-1 77.8%

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

      2. unsub-neg 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in re around 0 53.7%

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

   7. Taylor expanded in im around 0 54.2%

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

      1. *-commutative 75.7%

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

   9. Simplified 54.2%

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

   if 1.2599999999999999e158 < re

   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 11.1%

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

      1. count-2 11.1%

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

   6. Simplified 11.1%

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

   7. Taylor expanded in re around 0 32.8%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 32.8%

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

      2. unsub-neg 32.8%

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

   9. Simplified 32.8%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 32.8%

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

   11. Applied egg-rr 32.8%

       \[\leadsto 2 - \color{blue}{re \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.26 \cdot 10^{+158}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.3% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0009:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 0.0009) 1.0 (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0009) {
		tmp = 1.0;
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0009d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 - (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0009) {
		tmp = 1.0;
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0009:
		tmp = 1.0
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0009)
		tmp = 1.0;
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0009)
		tmp = 1.0;
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0009], 1.0, N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.9999999999999998e-4

   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 70.2%

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

      1. neg-mul-1 70.2%

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

      2. unsub-neg 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in re around 0 40.9%

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

   7. Taylor expanded in im around 0 40.5%

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

   if 8.9999999999999998e-4 < im

   1. Initial program 99.9%

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

   4. Applied egg-rr 3.7%

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

      1. count-2 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in re around 0 17.9%

      \[\leadsto \color{blue}{2 + -1 \cdot {re}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 17.9%

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

      2. unsub-neg 17.9%

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

   9. Simplified 17.9%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 17.9%

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

   11. Applied egg-rr 17.9%

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

Alternative 18: 28.2% 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 76.5%

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

   1. neg-mul-1 76.5%

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

   2. unsub-neg 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in re around 0 46.4%

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

7. Taylor expanded in im around 0 31.7%

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

Alternative 19: 3.8% 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 64.1%

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

5. Applied egg-rr 4.0%

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

   1. metadata-eval 4.0%

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

7. Applied egg-rr 4.0%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))