math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.7s

Alternatives: 22

Speedup: 1.0×

• 49.4% 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: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(t\_0 + 2\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 0.4)
     (*
      (* 0.5 (cos re))
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 6.5e+99)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* 0.5 (* (cos re) (- (+ t_0 2.0) im)))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 0.4) {
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 0.4d0) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((t_0 + 2.0d0) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 0.4) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 0.4:
		tmp = (0.5 * math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 6.5e+99:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * ((t_0 + 2.0) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 0.4)
		tmp = Float64(Float64(0.5 * cos(re)) * 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 <= 6.5e+99)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(t_0 + 2.0) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 0.4)
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 6.5e+99)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.4], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * 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, 6.5e+99], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$0 + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.40000000000000002

   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 87.4%

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

   4. Taylor expanded in im around 0 63.5%

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

      1. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   if 0.40000000000000002 < im < 6.5000000000000004e99

   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 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 0.1%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in re around inf 96.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% 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 73.8%

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

   1. neg-mul-1 73.8%

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

   2. unsub-neg 73.8%

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

5. Simplified 73.8%

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

6. Final simplification 73.8%

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

Alternative 4: 74.5% 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 73.8%

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

   1. neg-mul-1 73.8%

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

   2. unsub-neg 73.8%

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

5. Simplified 73.8%

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

6. Taylor expanded in im around 0 72.9%

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

7. Final simplification 72.9%

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

Alternative 5: 72.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 4.3:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(t\_0 + 2\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 4.3)
     (*
      (* 0.5 (cos re))
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 6.5e+99)
       (+ 0.5 (* 0.5 (exp im)))
       (* 0.5 (* (cos re) (- (+ t_0 2.0) im)))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.3) {
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 4.3d0) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((t_0 + 2.0d0) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.3) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 4.3:
		tmp = (0.5 * math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * ((t_0 + 2.0) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 4.3)
		tmp = Float64(Float64(0.5 * cos(re)) * 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 <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(t_0 + 2.0) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 4.3)
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.3], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * 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, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$0 + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.29999999999999982

   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 87.4%

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

   4. Taylor expanded in im around 0 63.5%

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

      1. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   if 4.29999999999999982 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 0.1%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in re around inf 96.2%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.3:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.1)
   (*
    (* 0.5 (cos re))
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (if (<= im 6.5e+99)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      0.5
      (*
       (cos re)
       (-
        (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0)
        im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.1) {
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.1d0) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (cos(re) * (((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.1) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.1:
		tmp = (0.5 * math.cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.1)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.1)
		tmp = (0.5 * cos(re)) * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.1], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * 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 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.10000000000000009

   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 87.4%

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

   4. Taylor expanded in im around 0 87.4%

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

   if 3.10000000000000009 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 0.1%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in re around inf 96.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(t\_0 + 2\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 4.2)
     (* (* 0.5 (cos re)) (+ (- 1.0 im) (+ 1.0 t_0)))
     (if (<= im 6.5e+99)
       (+ 0.5 (* 0.5 (exp im)))
       (* 0.5 (* (cos re) (- (+ t_0 2.0) im)))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.2) {
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 4.2d0) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 - im) + (1.0d0 + t_0))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((t_0 + 2.0d0) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 4.2) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 - im) + (1.0 + t_0));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((t_0 + 2.0) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 4.2:
		tmp = (0.5 * math.cos(re)) * ((1.0 - im) + (1.0 + t_0))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * ((t_0 + 2.0) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 4.2)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(1.0 - im) + Float64(1.0 + t_0)));
	elseif (im <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(t_0 + 2.0) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 4.2)
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + t_0));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.5 * (cos(re) * ((t_0 + 2.0) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.2], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$0 + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.20000000000000018

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.1%

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

      1. neg-mul-1 65.1%

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

      2. unsub-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 63.2%

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

      1. *-commutative 63.5%

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

   8. Simplified 63.2%

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

   if 4.20000000000000018 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 0.1%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in re around inf 96.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7)
   (* (* 0.5 (cos re)) (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (if (<= im 6.5e+99)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      0.5
      (*
       (cos re)
       (-
        (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0)
        im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.7d0) then
        tmp = (0.5d0 * cos(re)) * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = 0.5d0 * (cos(re) * (((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0) - im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = (0.5 * Math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.7:
		tmp = (0.5 * math.cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.7)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.7)
		tmp = (0.5 * cos(re)) * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = 0.5 * (cos(re) * (((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0) - im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.7], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * 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, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.1%

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

      1. neg-mul-1 65.1%

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

      2. unsub-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 79.9%

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

   if 2.7000000000000002 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 0.1%

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

   8. Simplified 96.2%

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

   9. Taylor expanded in re around inf 96.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right) - im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 2.2)
     (* t_0 (+ (- 1.0 im) (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 6.5e+99)
       (+ 0.5 (* 0.5 (exp im)))
       (*
        t_0
        (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 2.2) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 2.2d0) then
        tmp = t_0 * ((1.0d0 - im) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = t_0 * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 2.2) {
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 2.2:
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 2.2)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im) + Float64(1.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))));
	elseif (im <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(t_0 * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 2.2)
		tmp = t_0 * ((1.0 - im) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = t_0 * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.2], 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, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.2000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.1%

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

      1. neg-mul-1 65.1%

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

      2. unsub-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 79.9%

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

   if 2.2000000000000002 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 - im\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.85)
   (* 0.5 (* (cos re) (- (+ 2.0 (* im (+ 1.0 (* 0.5 im)))) im)))
   (if (<= im 6.5e+99)
     (+ 0.5 (* 0.5 (exp im)))
     (*
      (* 0.5 (cos re))
      (+ (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = (0.5 * cos(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.85d0) then
        tmp = 0.5d0 * (cos(re) * ((2.0d0 + (im * (1.0d0 + (0.5d0 * im)))) - im))
    else if (im <= 6.5d+99) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = (0.5d0 * cos(re)) * ((im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))) + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85) {
		tmp = 0.5 * (Math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	} else if (im <= 6.5e+99) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85:
		tmp = 0.5 * (math.cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im))
	elif im <= 6.5e+99:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))) - im)));
	elseif (im <= 6.5e+99)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))) + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85)
		tmp = 0.5 * (cos(re) * ((2.0 + (im * (1.0 + (0.5 * im)))) - im));
	elseif (im <= 6.5e+99)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = (0.5 * cos(re)) * ((im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 6.5e+99], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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 65.1%

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

      1. neg-mul-1 65.1%

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

      2. unsub-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 79.9%

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

   7. Taylor expanded in re around inf 79.9%

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

   if 1.8500000000000001 < im < 6.5000000000000004e99

   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 re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   9. Simplified 77.8%

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

   if 6.5000000000000004e99 < 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 96.2%

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

      1. *-commutative 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right) + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot \left(1 + 0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 2.9:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(t\_0 - im\right)\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+153}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))
   (if (<= im 2.9)
     (* 0.5 (* (cos re) (- t_0 im)))
     (if (<= im 9.2e+153) (+ 0.5 (* 0.5 (exp im))) (* (* 0.5 (cos re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * (1.0 + (0.5 * im)));
	double tmp;
	if (im <= 2.9) {
		tmp = 0.5 * (cos(re) * (t_0 - im));
	} else if (im <= 9.2e+153) {
		tmp = 0.5 + (0.5 * exp(im));
	} else {
		tmp = (0.5 * cos(re)) * 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 + (0.5d0 * im)))
    if (im <= 2.9d0) then
        tmp = 0.5d0 * (cos(re) * (t_0 - im))
    else if (im <= 9.2d+153) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    else
        tmp = (0.5d0 * cos(re)) * 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 + (0.5 * im)));
	double tmp;
	if (im <= 2.9) {
		tmp = 0.5 * (Math.cos(re) * (t_0 - im));
	} else if (im <= 9.2e+153) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} else {
		tmp = (0.5 * Math.cos(re)) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * (1.0 + (0.5 * im)))
	tmp = 0
	if im <= 2.9:
		tmp = 0.5 * (math.cos(re) * (t_0 - im))
	elif im <= 9.2e+153:
		tmp = 0.5 + (0.5 * math.exp(im))
	else:
		tmp = (0.5 * math.cos(re)) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * Float64(1.0 + Float64(0.5 * im))))
	tmp = 0.0
	if (im <= 2.9)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(t_0 - im)));
	elseif (im <= 9.2e+153)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * (1.0 + (0.5 * im)));
	tmp = 0.0;
	if (im <= 2.9)
		tmp = 0.5 * (cos(re) * (t_0 - im));
	elseif (im <= 9.2e+153)
		tmp = 0.5 + (0.5 * exp(im));
	else
		tmp = (0.5 * cos(re)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 2.9], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(t$95$0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+153], N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.89999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.1%

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

      1. neg-mul-1 65.1%

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

      2. unsub-neg 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in im around 0 79.9%

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

   7. Taylor expanded in re around inf 79.9%

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

   if 2.89999999999999991 < im < 9.2000000000000005e153

   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 re around 0 75.0%

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

      1. distribute-lft-in 75.0%

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

      2. metadata-eval 75.0%

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

   9. Simplified 75.0%

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

   if 9.2000000000000005e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

Alternative 12: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+153}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.0)
   (cos re)
   (if (<= im 9.2e+153)
     (+ 0.5 (* 0.5 (exp im)))
     (* (* 0.5 (cos re)) (+ 2.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.0) {
		tmp = cos(re);
	} else if (im <= 9.2e+153) {
		tmp = 0.5 + (0.5 * exp(im));
	} 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 <= 2.0d0) then
        tmp = cos(re)
    else if (im <= 9.2d+153) then
        tmp = 0.5d0 + (0.5d0 * exp(im))
    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 <= 2.0) {
		tmp = Math.cos(re);
	} else if (im <= 9.2e+153) {
		tmp = 0.5 + (0.5 * Math.exp(im));
	} 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 <= 2.0:
		tmp = math.cos(re)
	elif im <= 9.2e+153:
		tmp = 0.5 + (0.5 * math.exp(im))
	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 <= 2.0)
		tmp = cos(re);
	elseif (im <= 9.2e+153)
		tmp = Float64(0.5 + Float64(0.5 * exp(im)));
	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 <= 2.0)
		tmp = cos(re);
	elseif (im <= 9.2e+153)
		tmp = 0.5 + (0.5 * exp(im));
	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, 2.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 9.2e+153], N[(0.5 + N[(0.5 * N[Exp[im], $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 < 2

   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 64.3%

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

   if 2 < im < 9.2000000000000005e153

   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 re around 0 75.0%

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

      1. distribute-lft-in 75.0%

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

      2. metadata-eval 75.0%

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

   9. Simplified 75.0%

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

   if 9.2000000000000005e153 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

Alternative 13: 68.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.65:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.65) (cos re) (+ 0.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.65) {
		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 <= 2.65d0) 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 <= 2.65) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.65:
		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 <= 2.65)
		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 <= 2.65)
		tmp = cos(re);
	else
		tmp = 0.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.65], 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 < 2.64999999999999991

   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 64.3%

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

   if 2.64999999999999991 < 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 re around 0 70.3%

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

      1. distribute-lft-in 70.3%

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

      2. metadata-eval 70.3%

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

   9. Simplified 70.3%

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

Alternative 14: 63.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 26000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;2 - re \cdot re\\ \mathbf{else}:\\ \;\;\;\;1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 26000.0)
   (cos re)
   (if (<= im 5e+102)
     (- 2.0 (* re re))
     (+ 1.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 26000.0) {
		tmp = cos(re);
	} else if (im <= 5e+102) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 26000.0d0) then
        tmp = cos(re)
    else if (im <= 5d+102) then
        tmp = 2.0d0 - (re * re)
    else
        tmp = 1.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 26000.0) {
		tmp = Math.cos(re);
	} else if (im <= 5e+102) {
		tmp = 2.0 - (re * re);
	} else {
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 26000.0:
		tmp = math.cos(re)
	elif im <= 5e+102:
		tmp = 2.0 - (re * re)
	else:
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 26000.0)
		tmp = cos(re);
	elseif (im <= 5e+102)
		tmp = Float64(2.0 - Float64(re * re));
	else
		tmp = Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 26000.0)
		tmp = cos(re);
	elseif (im <= 5e+102)
		tmp = 2.0 - (re * re);
	else
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 26000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5e+102], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 26000

   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 64.0%

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

   if 26000 < im < 5e102

   1. Initial program 100.0%

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

   4. Applied egg-rr 3.1%

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

      1. count-2 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in re around 0 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%

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

   if 5e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

   7. Taylor expanded in re around 0 70.5%

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

      1. distribute-lft-in 70.5%

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

      2. metadata-eval 70.5%

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

   9. Simplified 70.5%

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

   10. Taylor expanded in im around 0 70.5%

       \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 70.5%

          \[\leadsto 1 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   12. Simplified 70.5%

       \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 44.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+195}:\\ \;\;\;\;1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.08e+195)
   (+ 1.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))
   (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.08e+195) {
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} 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 <= 1.08d+195) then
        tmp = 1.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    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 <= 1.08e+195) {
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.08e+195:
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.08e+195)
		tmp = Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.08e+195)
		tmp = 1.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.08e+195], N[(1.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.0800000000000001e195

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in im around 0 73.4%

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

   7. Taylor expanded in re around 0 45.3%

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

      1. distribute-lft-in 45.3%

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

      2. metadata-eval 45.3%

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

   9. Simplified 45.3%

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

   10. Taylor expanded in im around 0 44.3%

       \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 44.3%

          \[\leadsto 1 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   12. Simplified 44.3%

       \[\leadsto \color{blue}{1 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]

   if 1.0800000000000001e195 < 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 10.5%

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

      1. count-2 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in re around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. unsub-neg 30.3%

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

   9. Simplified 30.3%

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

       1. unpow2 30.3%

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

   11. Applied egg-rr 30.3%

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

Alternative 16: 48.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+195}:\\ \;\;\;\;0.5 \cdot \left(\left(2 + im \cdot \left(1 + 0.5 \cdot im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.08e+195)
   (* 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 <= 1.08e+195) {
		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 <= 1.08d+195) 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 <= 1.08e+195) {
		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 <= 1.08e+195:
		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 <= 1.08e+195)
		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 <= 1.08e+195)
		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, 1.08e+195], 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 < 1.0800000000000001e195

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in im around 0 74.3%

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

   7. Taylor expanded in re around 0 46.3%

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

   if 1.0800000000000001e195 < 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 10.5%

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

      1. count-2 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in re around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. unsub-neg 30.3%

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

   9. Simplified 30.3%

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

       1. unpow2 30.3%

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

   11. Applied egg-rr 30.3%

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

Alternative 17: 47.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+195}:\\ \;\;\;\;1 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.08e+195) (+ 1.0 (* im (+ 0.5 (* im 0.25)))) (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.08e+195) {
		tmp = 1.0 + (im * (0.5 + (im * 0.25)));
	} 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 <= 1.08d+195) then
        tmp = 1.0d0 + (im * (0.5d0 + (im * 0.25d0)))
    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 <= 1.08e+195) {
		tmp = 1.0 + (im * (0.5 + (im * 0.25)));
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.08e+195:
		tmp = 1.0 + (im * (0.5 + (im * 0.25)))
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.08e+195)
		tmp = Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25))));
	else
		tmp = Float64(2.0 - Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.08e+195)
		tmp = 1.0 + (im * (0.5 + (im * 0.25)));
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.08e+195], N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.0800000000000001e195

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 74.3%

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

      1. neg-mul-1 74.3%

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

      2. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in im around 0 73.4%

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

   7. Taylor expanded in re around 0 45.3%

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

      1. distribute-lft-in 45.3%

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

      2. metadata-eval 45.3%

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

   9. Simplified 45.3%

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

   10. Taylor expanded in im around 0 46.1%

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

       1. *-commutative 46.1%

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

   12. Simplified 46.1%

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

   if 1.0800000000000001e195 < 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 10.5%

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

      1. count-2 10.5%

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

   6. Simplified 10.5%

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

   7. Taylor expanded in re around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. unsub-neg 30.3%

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

   9. Simplified 30.3%

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

       1. unpow2 30.3%

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

   11. Applied egg-rr 30.3%

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

Alternative 18: 32.5% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 26000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 - re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 26000.0) 1.0 (- 2.0 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 26000.0) {
		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 <= 26000.0d0) 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 <= 26000.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 - (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 26000.0:
		tmp = 1.0
	else:
		tmp = 2.0 - (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 26000.0)
		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 <= 26000.0)
		tmp = 1.0;
	else
		tmp = 2.0 - (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 26000.0], 1.0, N[(2.0 - N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 26000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 65.3%

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

      1. neg-mul-1 65.3%

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

      2. unsub-neg 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in im around 0 64.1%

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

   7. Taylor expanded in re around 0 34.9%

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

      1. distribute-lft-in 34.9%

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

      2. metadata-eval 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in im around 0 34.6%

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

   if 26000 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 3.1%

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

      1. count-2 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in re around 0 15.2%

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

      1. mul-1-neg 15.2%

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

      2. unsub-neg 15.2%

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

   9. Simplified 15.2%

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

       1. unpow2 15.2%

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

   11. Applied egg-rr 15.2%

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

Alternative 19: 28.9% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 73.8%

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

   1. neg-mul-1 73.8%

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

   2. unsub-neg 73.8%

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

5. Simplified 73.8%

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

6. Taylor expanded in im around 0 72.9%

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

7. Taylor expanded in re around 0 43.5%

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

   1. distribute-lft-in 43.5%

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

   2. metadata-eval 43.5%

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

9. Simplified 43.5%

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

10. Taylor expanded in im around 0 26.7%

    \[\leadsto \color{blue}{1 + 0.5 \cdot im} \]
11. Step-by-step derivation

    1. +-commutative 26.7%

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

    2. *-commutative 26.7%

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

12. Simplified 26.7%

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

13. Final simplification 26.7%

    \[\leadsto 1 + 0.5 \cdot im \]
14. Add Preprocessing

Alternative 20: 29.1% 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 73.8%

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

   1. neg-mul-1 73.8%

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

   2. unsub-neg 73.8%

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

5. Simplified 73.8%

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

6. Taylor expanded in im around 0 72.9%

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

7. Taylor expanded in re around 0 43.5%

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

   1. distribute-lft-in 43.5%

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

   2. metadata-eval 43.5%

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

9. Simplified 43.5%

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

10. Taylor expanded in im around 0 26.6%

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

Alternative 21: 9.3% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.75)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.75

Julia

function code(re, im)
	return 0.75
end

MATLAB

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

Wolfram

code[re_, im_] := 0.75

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 62.1%

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

5. Applied egg-rr 8.8%

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

   1. metadata-eval 8.8%

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

7. Applied egg-rr 8.8%

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

Alternative 22: 2.4% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 2.3%

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

   1. pow-base-1 2.3%

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

   2. metadata-eval 2.3%

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

6. Simplified 2.3%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))