math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 20

Speedup: 1.0×

• 49.2% 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: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im)) (exp im)) 4.0)
     (*
      t_0
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (* t_0 (+ (exp im) 3.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((exp(-im) + exp(im)) <= 4.0) {
		tmp = t_0 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (exp(im) + 3.0);
	}
	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 ((exp(-im) + exp(im)) <= 4.0d0) then
        tmp = t_0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (exp(im) + 3.0d0)
    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 ((Math.exp(-im) + Math.exp(im)) <= 4.0) {
		tmp = t_0 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if (math.exp(-im) + math.exp(im)) <= 4.0:
		tmp = t_0 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im)) + exp(im)) <= 4.0)
		tmp = Float64(t_0 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if ((exp(-im) + exp(im)) <= 4.0)
		tmp = t_0 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision], 4.0], N[(t$95$0 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 99.6%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 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 50.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im)) (exp im)) 4.0)
     (*
      t_0
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
     (* t_0 (+ (exp im) 3.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((exp(-im) + exp(im)) <= 4.0) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = t_0 * (exp(im) + 3.0);
	}
	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 ((exp(-im) + exp(im)) <= 4.0d0) then
        tmp = t_0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else
        tmp = t_0 * (exp(im) + 3.0d0)
    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 ((Math.exp(-im) + Math.exp(im)) <= 4.0) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else {
		tmp = t_0 * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if (math.exp(-im) + math.exp(im)) <= 4.0:
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	else:
		tmp = t_0 * (math.exp(im) + 3.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(exp(Float64(-im)) + exp(im)) <= 4.0)
		tmp = Float64(t_0 * 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(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if ((exp(-im) + exp(im)) <= 4.0)
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	else
		tmp = t_0 * (exp(im) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision], 4.0], N[(t$95$0 * 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[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 99.6%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   if 4 < (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 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 50.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;t\_1 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(4 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (cos re))))
   (if (<= im 0.0065)
     (*
      t_1
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 1e+103)
       (* 0.5 (+ (exp (- im)) (exp im)))
       (* t_1 (+ 4.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * cos(re);
	double tmp;
	if (im <= 0.0065) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = t_1 * (4.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * cos(re)
    if (im <= 0.0065d0) then
        tmp = t_1 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = t_1 * (4.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 0.0065) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = t_1 * (4.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 0.0065:
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = t_1 * (4.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 0.0065)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + t_0)));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_1 * Float64(4.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 0.0065)
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = t_1 * (4.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0065], N[(t$95$1 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(4.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0064999999999999997

   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 90.3%

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

   4. Taylor expanded in im around 0 69.3%

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

      1. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   if 0.0064999999999999997 < im < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 55.6%

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

   if 1e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;t\_1 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(4 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (cos re))))
   (if (<= im 5.2)
     (*
      t_1
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 1e+103) (+ (* 0.5 (exp im)) 1.5) (* t_1 (+ 4.0 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * cos(re);
	double tmp;
	if (im <= 5.2) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = t_1 * (4.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * cos(re)
    if (im <= 5.2d0) then
        tmp = t_1 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 1d+103) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = t_1 * (4.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double t_1 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 5.2) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = t_1 * (4.0 + t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	t_1 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 5.2:
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 1e+103:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = t_1 * (4.0 + t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 5.2)
		tmp = Float64(t_1 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0))) + Float64(1.0 + t_0)));
	elseif (im <= 1e+103)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(t_1 * Float64(4.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	t_1 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 5.2)
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 1e+103)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = t_1 * (4.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5.2], N[(t$95$1 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(t$95$1 * N[(4.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   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 90.3%

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

   4. Taylor expanded in im around 0 69.3%

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

      1. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   if 5.20000000000000018 < im < 1e103

   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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 55.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 55.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 55.6%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 55.6%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 55.6%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 1e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + t\_1\right) + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
   (if (<= im 0.0065)
     (* t_0 (+ (+ 1.0 t_1) (+ 1.0 (* im (+ (* 0.5 im) -1.0)))))
     (if (<= im 1e+103)
       (*
        0.5
        (+
         (exp im)
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
       (* t_0 (+ 4.0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 0.0065) {
		tmp = t_0 * ((1.0 + t_1) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (4.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    if (im <= 0.0065d0) then
        tmp = t_0 * ((1.0d0 + t_1) + (1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = t_0 * (4.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	double tmp;
	if (im <= 0.0065) {
		tmp = t_0 * ((1.0 + t_1) + (1.0 + (im * ((0.5 * im) + -1.0))));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = t_0 * (4.0 + t_1);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))
	tmp = 0
	if im <= 0.0065:
		tmp = t_0 * ((1.0 + t_1) + (1.0 + (im * ((0.5 * im) + -1.0))))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = t_0 * (4.0 + t_1)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))
	tmp = 0.0
	if (im <= 0.0065)
		tmp = Float64(t_0 * Float64(Float64(1.0 + t_1) + Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0)))));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(t_0 * Float64(4.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))));
	tmp = 0.0;
	if (im <= 0.0065)
		tmp = t_0 * ((1.0 + t_1) + (1.0 + (im * ((0.5 * im) + -1.0))));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = t_0 * (4.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.0065], N[(t$95$0 * N[(N[(1.0 + t$95$1), $MachinePrecision] + N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(4.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0064999999999999997

   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 90.3%

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

   4. Taylor expanded in im around 0 69.3%

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

      1. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in im around 0 68.9%

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

   if 0.0064999999999999997 < im < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 55.6%

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

   if 1e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0065:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) + \left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.000116:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.000116)
   (cos re)
   (if (<= im 1e+103)
     (*
      0.5
      (+
       (exp im)
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
     (*
      (* 0.5 (cos re))
      (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.000116) {
		tmp = cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.000116d0) then
        tmp = cos(re)
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.000116) {
		tmp = Math.cos(re);
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.000116:
		tmp = math.cos(re)
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.000116)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.000116)
		tmp = cos(re);
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + (1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.000116], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.16e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. fma-define 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around 0 69.6%

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

   if 1.16e-4 < im < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in re around 0 55.6%

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

   if 1e103 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 100.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.000116:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.9:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.9)
   (cos re)
   (if (<= im 5.7e+153)
     (+ (* 0.5 (exp im)) 1.5)
     (* (* 0.5 (cos re)) (+ 4.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.9) {
		tmp = cos(re);
	} else if (im <= 5.7e+153) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = (0.5 * cos(re)) * (4.0 + (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 <= 3.9d0) then
        tmp = cos(re)
    else if (im <= 5.7d+153) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = (0.5d0 * cos(re)) * (4.0d0 + (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 <= 3.9) {
		tmp = Math.cos(re);
	} else if (im <= 5.7e+153) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = (0.5 * Math.cos(re)) * (4.0 + (im * (1.0 + (0.5 * im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.9:
		tmp = math.cos(re)
	elif im <= 5.7e+153:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = (0.5 * math.cos(re)) * (4.0 + (im * (1.0 + (0.5 * im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.9)
		tmp = cos(re);
	elseif (im <= 5.7e+153)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(4.0 + 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 <= 3.9)
		tmp = cos(re);
	elseif (im <= 5.7e+153)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = (0.5 * cos(re)) * (4.0 + (im * (1.0 + (0.5 * im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 5.7e+153], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.89999999999999991

   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 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. fma-define 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around 0 69.6%

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

   if 3.89999999999999991 < im < 5.69999999999999987e153

   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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 67.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 67.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 67.8%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 67.8%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 67.8%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   if 5.69999999999999987e153 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in im around 0 100.0%

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

Alternative 9: 69.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6) (cos re) (+ (* 0.5 (exp im)) 1.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * exp(im)) + 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.6d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * exp(im)) + 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6:
		tmp = math.cos(re)
	else:
		tmp = (0.5 * math.exp(im)) + 1.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6)
		tmp = cos(re);
	else
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im)) + 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. unpow2 86.0%

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

      3. fma-define 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around 0 69.6%

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

   if 3.60000000000000009 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 70.2%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 70.2%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 70.2%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.5e+33)
   (cos re)
   (if (<= im 6.2e+100)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.5e+33) {
		tmp = cos(re);
	} else if (im <= 6.2e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 4.5d+33) then
        tmp = cos(re)
    else if (im <= 6.2d+100) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 4.5e+33) {
		tmp = Math.cos(re);
	} else if (im <= 6.2e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.5e+33:
		tmp = math.cos(re)
	elif im <= 6.2e+100:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.5e+33)
		tmp = cos(re);
	elseif (im <= 6.2e+100)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 4.5e+33)
		tmp = cos(re);
	elseif (im <= 6.2e+100)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.5e+33], N[Cos[re], $MachinePrecision], If[LessEqual[im, 6.2e+100], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 4.5e33

   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 82.8%

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

      1. +-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. fma-define 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in im around 0 67.0%

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

   if 4.5e33 < im < 6.20000000000000014e100

   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 4.5%

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

      1. +-commutative 4.5%

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

      2. unpow2 4.5%

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

      3. fma-define 4.5%

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

   5. Simplified 4.5%

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

   7. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 2.3%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 2.3%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 56.2%

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

       1. unpow2 56.2%

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

   12. Applied egg-rr 56.2%

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

   if 6.20000000000000014e100 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 41.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+29)
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))
   (if (<= im 9.2e+101)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else if (im <= 9.2e+101) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 1.5d+29) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    else if (im <= 9.2d+101) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 1.5e+29) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	} else if (im <= 9.2e+101) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+29:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	elif im <= 9.2e+101:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+29)
		tmp = Float64(0.5 * 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(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	elseif (im <= 9.2e+101)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 1.5e+29)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	elseif (im <= 9.2e+101)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+29], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.2e+101], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 1.5e29

   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 90.5%

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

   4. Taylor expanded in re around 0 52.1%

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

   5. Taylor expanded in im around 0 37.2%

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

      1. *-commutative 67.7%

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

   7. Simplified 37.2%

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

   if 1.5e29 < im < 9.2000000000000005e101

   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 4.3%

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

      1. +-commutative 4.3%

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

      2. unpow2 4.3%

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

      3. fma-define 4.3%

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

   5. Simplified 4.3%

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

   7. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 2.1%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 2.1%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 42.5%

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

       1. unpow2 42.5%

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

   12. Applied egg-rr 42.5%

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

   if 9.2000000000000005e101 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 185:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 185.0)
   (*
    0.5
    (+
     (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
     (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
   (if (<= im 3.55e+101)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 185.0) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 3.55e+101) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 185.0d0) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 3.55d+101) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 185.0) {
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 3.55e+101) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 185.0:
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 3.55e+101:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 185.0)
		tmp = Float64(0.5 * 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 <= 3.55e+101)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 185.0)
		tmp = 0.5 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 3.55e+101)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 185.0], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.55e+101], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 185

   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 90.3%

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

   4. Taylor expanded in re around 0 51.2%

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

   5. Taylor expanded in im around 0 51.0%

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

   if 185 < im < 3.55000000000000026e101

   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 4.1%

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

      1. +-commutative 4.1%

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

      2. unpow2 4.1%

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

      3. fma-define 4.1%

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

   5. Simplified 4.1%

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

   7. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 1.7%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 1.7%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 32.0%

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

       1. unpow2 32.0%

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

   12. Applied egg-rr 32.0%

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

   if 3.55000000000000026e101 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 185:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+101}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot 0.5277777777777778 + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.2e+30)
   (* 0.5 (+ (+ 1.0 (* im (+ (* im 0.5277777777777778) -1.0))) (+ im 1.0)))
   (if (<= im 5.3e+100)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.2e+30) {
		tmp = 0.5 * ((1.0 + (im * ((im * 0.5277777777777778) + -1.0))) + (im + 1.0));
	} else if (im <= 5.3e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 9.2d+30) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((im * 0.5277777777777778d0) + (-1.0d0)))) + (im + 1.0d0))
    else if (im <= 5.3d+100) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 9.2e+30) {
		tmp = 0.5 * ((1.0 + (im * ((im * 0.5277777777777778) + -1.0))) + (im + 1.0));
	} else if (im <= 5.3e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 9.2e+30:
		tmp = 0.5 * ((1.0 + (im * ((im * 0.5277777777777778) + -1.0))) + (im + 1.0))
	elif im <= 5.3e+100:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 9.2e+30)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(im * 0.5277777777777778) + -1.0))) + Float64(im + 1.0)));
	elseif (im <= 5.3e+100)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 9.2e+30)
		tmp = 0.5 * ((1.0 + (im * ((im * 0.5277777777777778) + -1.0))) + (im + 1.0));
	elseif (im <= 5.3e+100)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.2e+30], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(im * 0.5277777777777778), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.3e+100], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 9.2e30

   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 90.6%

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

   4. Taylor expanded in re around 0 52.1%

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

   5. Taylor expanded in im around 0 49.4%

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

   7. Applied egg-rr 47.1%

      \[\leadsto 0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + \color{blue}{0.027777777777777776}\right) - 1\right)\right) + \left(1 + im\right)\right) \]

   if 9.2e30 < im < 5.2999999999999998e100

   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 4.4%

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

      1. +-commutative 4.4%

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

      2. unpow2 4.4%

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

      3. fma-define 4.4%

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

   5. Simplified 4.4%

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

   7. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 2.1%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 2.1%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 50.6%

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

       1. unpow2 50.6%

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

   12. Applied egg-rr 50.6%

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

   if 5.2999999999999998e100 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.2 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(im \cdot 0.5277777777777778 + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 5.3 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+32)
   (* 0.5 (+ (+ 1.0 (* im (+ (* 0.5 im) -1.0))) (+ im 1.0)))
   (if (<= im 5.2e+100)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+32) {
		tmp = 0.5 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	} else if (im <= 5.2e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 1d+32) then
        tmp = 0.5d0 * ((1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))) + (im + 1.0d0))
    else if (im <= 5.2d+100) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 1e+32) {
		tmp = 0.5 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	} else if (im <= 5.2e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+32:
		tmp = 0.5 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0))
	elif im <= 5.2e+100:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+32)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0))) + Float64(im + 1.0)));
	elseif (im <= 5.2e+100)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 1e+32)
		tmp = 0.5 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	elseif (im <= 5.2e+100)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+32], N[(0.5 * N[(N[(1.0 + N[(im * N[(N[(0.5 * im), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.2e+100], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 1.00000000000000005e32

   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 90.6%

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

   4. Taylor expanded in re around 0 52.1%

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

   5. Taylor expanded in im around 0 49.4%

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

   6. Taylor expanded in im around 0 47.1%

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

   if 1.00000000000000005e32 < im < 5.2000000000000003e100

   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 4.4%

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

      1. +-commutative 4.4%

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

      2. unpow2 4.4%

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

      3. fma-define 4.4%

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

   5. Simplified 4.4%

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

   7. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 2.1%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 2.1%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 50.6%

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

       1. unpow2 50.6%

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

   12. Applied egg-rr 50.6%

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

   if 5.2000000000000003e100 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10^{+32}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+100}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+29)
   1.0
   (if (<= im 1.7e+100)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 1.0;
	} else if (im <= 1.7e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (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 <= 1.5d+29) then
        tmp = 1.0d0
    else if (im <= 1.7d+100) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (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 <= 1.5e+29) {
		tmp = 1.0;
	} else if (im <= 1.7e+100) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+29:
		tmp = 1.0
	elif im <= 1.7e+100:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+29)
		tmp = 1.0;
	elseif (im <= 1.7e+100)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + 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 <= 1.5e+29)
		tmp = 1.0;
	elseif (im <= 1.7e+100)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+29], 1.0, If[LessEqual[im, 1.7e+100], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + 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 < 1.5e29

   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 68.0%

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

   4. Taylor expanded in re around 0 37.4%

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

   if 1.5e29 < im < 1.69999999999999997e100

   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 4.3%

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

      1. +-commutative 4.3%

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

      2. unpow2 4.3%

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

      3. fma-define 4.3%

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

   5. Simplified 4.3%

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

   7. Applied egg-rr 2.1%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 2.1%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 2.1%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 42.5%

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

       1. unpow2 42.5%

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

   12. Applied egg-rr 42.5%

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

   if 1.69999999999999997e100 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.5%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 75.4%

      \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 75.4%

         \[\leadsto 2 + im \cdot \left(0.5 + im \cdot \left(0.25 + \color{blue}{im \cdot 0.08333333333333333}\right)\right) \]

   10. Simplified 75.4%

       \[\leadsto \color{blue}{2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 39.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+117}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+29)
   1.0
   (if (<= im 6.8e+117)
     (- (* -0.5 (* re re)) 2.0)
     (+ 2.0 (* im (+ 0.5 (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 1.0;
	} else if (im <= 6.8e+117) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.5d+29) then
        tmp = 1.0d0
    else if (im <= 6.8d+117) then
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 1.0;
	} else if (im <= 6.8e+117) {
		tmp = (-0.5 * (re * re)) - 2.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+29:
		tmp = 1.0
	elif im <= 6.8e+117:
		tmp = (-0.5 * (re * re)) - 2.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+29)
		tmp = 1.0;
	elseif (im <= 6.8e+117)
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e+29)
		tmp = 1.0;
	elseif (im <= 6.8e+117)
		tmp = (-0.5 * (re * re)) - 2.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+29], 1.0, If[LessEqual[im, 6.8e+117], N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.5e29

   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 68.0%

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

   4. Taylor expanded in re around 0 37.4%

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

   if 1.5e29 < im < 6.8000000000000002e117

   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 4.8%

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

      1. +-commutative 4.8%

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

      2. unpow2 4.8%

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

      3. fma-define 4.8%

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

   5. Simplified 4.8%

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

   7. Applied egg-rr 1.8%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 1.8%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 1.8%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 36.0%

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

       1. unpow2 36.0%

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

   12. Applied egg-rr 36.0%

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

   if 6.8000000000000002e117 < 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 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   5. Taylor expanded in re around 0 77.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(3 + e^{im}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 77.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{im} + 3\right)} \]

      2. distribute-lft-in 77.1%

         \[\leadsto \color{blue}{0.5 \cdot e^{im} + 0.5 \cdot 3} \]

      3. metadata-eval 77.1%

         \[\leadsto 0.5 \cdot e^{im} + \color{blue}{1.5} \]

   7. Simplified 77.1%

      \[\leadsto \color{blue}{0.5 \cdot e^{im} + 1.5} \]

   8. Taylor expanded in im around 0 56.1%

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

      1. *-commutative 56.1%

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

   10. Simplified 56.1%

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

Alternative 17: 31.7% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right) - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.5e+29) 1.0 (- (* -0.5 (* re re)) 2.0)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 1.0;
	} else {
		tmp = (-0.5 * (re * re)) - 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 <= 1.5d+29) then
        tmp = 1.0d0
    else
        tmp = ((-0.5d0) * (re * re)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.5e+29) {
		tmp = 1.0;
	} else {
		tmp = (-0.5 * (re * re)) - 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.5e+29:
		tmp = 1.0
	else:
		tmp = (-0.5 * (re * re)) - 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.5e+29)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-0.5 * Float64(re * re)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.5e+29)
		tmp = 1.0;
	else
		tmp = (-0.5 * (re * re)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.5e+29], 1.0, N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.5e29

   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 68.0%

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

   4. Taylor expanded in re around 0 37.4%

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

   if 1.5e29 < 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 53.0%

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

      1. +-commutative 53.0%

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

      2. unpow2 53.0%

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

      3. fma-define 53.0%

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

   5. Simplified 53.0%

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

   7. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{-3 + \cos re} \]
   8. Step-by-step derivation

      1. +-commutative 1.4%

         \[\leadsto \color{blue}{\cos re + -3} \]

   9. Simplified 1.4%

      \[\leadsto \color{blue}{\cos re + -3} \]

   10. Taylor expanded in re around 0 21.9%

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

       1. unpow2 21.9%

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

   12. Applied egg-rr 21.9%

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

Alternative 18: 29.4% 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 54.8%

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

4. Taylor expanded in re around 0 30.3%

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

Alternative 19: 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 61.9%

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

5. Applied egg-rr 9.6%

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

   1. metadata-eval 9.6%

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

7. Applied egg-rr 9.6%

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

Alternative 20: 3.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -2.0)

C

double code(double re, double im) {
	return -2.0;
}

Fortran

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

Java

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

Python

def code(re, im):
	return -2.0

Julia

function code(re, im)
	return -2.0
end

MATLAB

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

Wolfram

code[re_, im_] := -2.0

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 77.7%

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

   1. +-commutative 77.7%

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

   2. unpow2 77.7%

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

   3. fma-define 77.7%

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

5. Simplified 77.7%

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

7. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{-3 + \cos re} \]
8. Step-by-step derivation

   1. +-commutative 3.5%

      \[\leadsto \color{blue}{\cos re + -3} \]

9. Simplified 3.5%

   \[\leadsto \color{blue}{\cos re + -3} \]

10. Taylor expanded in re around 0 3.7%

    \[\leadsto \color{blue}{-2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))