math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 15

Speedup: 1.0×

• 49.5% 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: 75.4% 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 99.9%

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

   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 52.4%

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

4. Final simplification 76.9%

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

Alternative 3: 75.4% 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 99.9%

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

   4. Taylor expanded in im around 0 99.9%

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

      1. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   if 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 52.4%

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

4. Final simplification 76.9%

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

Alternative 4: 72.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ t_2 := 1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{if}\;im \leq 7.4:\\ \;\;\;\;t\_1 \cdot \left(t\_2 + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(4 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))
        (t_1 (* 0.5 (cos re)))
        (t_2
         (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 7.4)
     (* t_1 (+ t_2 (+ 1.0 t_0)))
     (if (<= im 1e+103) (* 0.5 (+ (exp im) t_2)) (* 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 t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 7.4) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (exp(im) + t_2);
	} 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) :: t_2
    real(8) :: tmp
    t_0 = im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))
    t_1 = 0.5d0 * cos(re)
    t_2 = 1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))
    if (im <= 7.4d0) then
        tmp = t_1 * (t_2 + (1.0d0 + t_0))
    else if (im <= 1d+103) then
        tmp = 0.5d0 * (exp(im) + t_2)
    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 t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	double tmp;
	if (im <= 7.4) {
		tmp = t_1 * (t_2 + (1.0 + t_0));
	} else if (im <= 1e+103) {
		tmp = 0.5 * (Math.exp(im) + t_2);
	} 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)
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))
	tmp = 0
	if im <= 7.4:
		tmp = t_1 * (t_2 + (1.0 + t_0))
	elif im <= 1e+103:
		tmp = 0.5 * (math.exp(im) + t_2)
	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))
	t_2 = Float64(1.0 + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))
	tmp = 0.0
	if (im <= 7.4)
		tmp = Float64(t_1 * Float64(t_2 + Float64(1.0 + t_0)));
	elseif (im <= 1e+103)
		tmp = Float64(0.5 * Float64(exp(im) + t_2));
	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);
	t_2 = 1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0));
	tmp = 0.0;
	if (im <= 7.4)
		tmp = t_1 * (t_2 + (1.0 + t_0));
	elseif (im <= 1e+103)
		tmp = 0.5 * (exp(im) + t_2);
	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]}, Block[{t$95$2 = N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 7.4], N[(t$95$1 * N[(t$95$2 + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(4.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < 7.4000000000000004

   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 84.3%

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

   4. Taylor expanded in im around 0 68.7%

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

      1. *-commutative 68.7%

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

   6. Simplified 68.7%

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

   if 7.4000000000000004 < 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 92.3%

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

   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 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7.4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(2 + \left(im + im \cdot \left(im + -1\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5)
   (* 0.5 (* (cos re) (+ 2.0 (+ im (* im (+ im -1.0))))))
   (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 <= 6.5) {
		tmp = 0.5 * (cos(re) * (2.0 + (im + (im * (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 = (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 <= 6.5d0) then
        tmp = 0.5d0 * (cos(re) * (2.0d0 + (im + (im * (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 = (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 <= 6.5) {
		tmp = 0.5 * (Math.cos(re) * (2.0 + (im + (im * (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 = (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 <= 6.5:
		tmp = 0.5 * (math.cos(re) * (2.0 + (im + (im * (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 = (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 <= 6.5)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(2.0 + Float64(im + Float64(im * Float64(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(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 <= 6.5)
		tmp = 0.5 * (cos(re) * (2.0 + (im + (im * (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 = (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, 6.5], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im + N[(im * N[(im + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+103], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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 < 6.5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.3%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in re around inf 80.2%

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

   if 6.5 < 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 92.3%

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

   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 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(2 + \left(im + im \cdot \left(im + -1\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(2 + \left(im + im \cdot \left(im + -1\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.0)
   (* 0.5 (* (cos re) (+ 2.0 (+ im (* im (+ im -1.0))))))
   (if (<= im 2.7e+154)
     (* 0.5 (+ (exp im) (- 1.0 im)))
     (* (cos re) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.0) {
		tmp = 0.5 * (cos(re) * (2.0 + (im + (im * (im + -1.0)))));
	} else if (im <= 2.7e+154) {
		tmp = 0.5 * (exp(im) + (1.0 - im));
	} else {
		tmp = cos(re) * (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 <= 5.0d0) then
        tmp = 0.5d0 * (cos(re) * (2.0d0 + (im + (im * (im + (-1.0d0))))))
    else if (im <= 2.7d+154) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 - im))
    else
        tmp = cos(re) * (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 <= 5.0) {
		tmp = 0.5 * (Math.cos(re) * (2.0 + (im + (im * (im + -1.0)))));
	} else if (im <= 2.7e+154) {
		tmp = 0.5 * (Math.exp(im) + (1.0 - im));
	} else {
		tmp = Math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.0:
		tmp = 0.5 * (math.cos(re) * (2.0 + (im + (im * (im + -1.0)))))
	elif im <= 2.7e+154:
		tmp = 0.5 * (math.exp(im) + (1.0 - im))
	else:
		tmp = math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(2.0 + Float64(im + Float64(im * Float64(im + -1.0))))));
	elseif (im <= 2.7e+154)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 - im)));
	else
		tmp = Float64(cos(re) * 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 <= 5.0)
		tmp = 0.5 * (cos(re) * (2.0 + (im + (im * (im + -1.0)))));
	elseif (im <= 2.7e+154)
		tmp = 0.5 * (exp(im) + (1.0 - im));
	else
		tmp = cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im + N[(im * N[(im + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 5

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.3%

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

   4. Taylor expanded in im around 0 84.1%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in re around inf 80.2%

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

   if 5 < im < 2.70000000000000006e154

   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 93.9%

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

   4. Taylor expanded in im around 0 93.9%

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

      1. neg-mul-1 93.9%

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

      2. unsub-neg 93.9%

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

   6. Simplified 93.9%

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

   if 2.70000000000000006e154 < 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%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(2 + \left(im + im \cdot \left(im + -1\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.7)
   (cos re)
   (if (<= im 2.7e+154)
     (* 0.5 (+ (exp im) (- 1.0 im)))
     (* (cos re) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.7) {
		tmp = cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 0.5 * (exp(im) + (1.0 - im));
	} else {
		tmp = cos(re) * (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 <= 1.7d0) then
        tmp = cos(re)
    else if (im <= 2.7d+154) then
        tmp = 0.5d0 * (exp(im) + (1.0d0 - im))
    else
        tmp = cos(re) * (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 <= 1.7) {
		tmp = Math.cos(re);
	} else if (im <= 2.7e+154) {
		tmp = 0.5 * (Math.exp(im) + (1.0 - im));
	} else {
		tmp = Math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.7:
		tmp = math.cos(re)
	elif im <= 2.7e+154:
		tmp = 0.5 * (math.exp(im) + (1.0 - im))
	else:
		tmp = math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.7)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = Float64(0.5 * Float64(exp(im) + Float64(1.0 - im)));
	else
		tmp = Float64(cos(re) * 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 <= 1.7)
		tmp = cos(re);
	elseif (im <= 2.7e+154)
		tmp = 0.5 * (exp(im) + (1.0 - im));
	else
		tmp = cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + N[(1.0 - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.2%

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

   if 1.69999999999999996 < im < 2.70000000000000006e154

   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 93.9%

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

   4. Taylor expanded in im around 0 93.9%

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

      1. neg-mul-1 93.9%

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

      2. unsub-neg 93.9%

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

   6. Simplified 93.9%

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

   if 2.70000000000000006e154 < 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%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + \left(1 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% 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 69.2%

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

   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 79.4%

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

      1. +-commutative 79.4%

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

      2. distribute-lft-in 79.4%

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

      3. metadata-eval 79.4%

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

   7. Simplified 79.4%

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

Alternative 9: 68.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6) (cos re) (* 0.5 (+ (exp im) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (exp(im) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (exp(im) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (Math.exp(im) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (math.exp(im) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(exp(im) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6)
		tmp = cos(re);
	else
		tmp = 0.5 * (exp(im) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.60000000000000009

   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 69.2%

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

   if 2.60000000000000009 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 79.4%

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

   4. Taylor expanded in im around 0 79.4%

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

      1. neg-mul-1 79.4%

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

      2. unsub-neg 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in im around 0 79.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{im} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 8:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 8.0)
   (cos re)
   (*
    0.5
    (+ 1.0 (+ 1.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 8.0d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * (1.0d0 + (1.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 8.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * (1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 8.0:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * (1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 8.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(1.0 + Float64(1.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 8.0)
		tmp = cos(re);
	else
		tmp = 0.5 * (1.0 + (1.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 8.0], N[Cos[re], $MachinePrecision], N[(0.5 * N[(1.0 + N[(1.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 69.2%

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

   if 8 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 79.4%

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

   4. Taylor expanded in im around 0 79.4%

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

      1. neg-mul-1 79.4%

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

      2. unsub-neg 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in im around 0 43.5%

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

      1. *-commutative 0.7%

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

   9. Simplified 43.5%

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

   10. Taylor expanded in im around 0 43.5%

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

Alternative 11: 44.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 66.7%

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

4. Taylor expanded in im around 0 48.4%

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

   1. neg-mul-1 48.4%

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

   2. unsub-neg 48.4%

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

6. Simplified 48.4%

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

7. Taylor expanded in im around 0 41.1%

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

   1. *-commutative 52.0%

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

9. Simplified 41.1%

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

10. Final simplification 41.1%

    \[\leadsto 0.5 \cdot \left(\left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right) + \left(1 - im\right)\right) \]
11. Add Preprocessing

Alternative 12: 44.2% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 66.7%

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

4. Taylor expanded in im around 0 48.4%

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

   1. neg-mul-1 48.4%

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

   2. unsub-neg 48.4%

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

6. Simplified 48.4%

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

7. Taylor expanded in im around 0 41.1%

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

   1. *-commutative 52.0%

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

9. Simplified 41.1%

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

10. Taylor expanded in im around 0 40.8%

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

Alternative 13: 47.3% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 73.7%

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

4. Taylor expanded in im around 0 63.4%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in re around 0 43.2%

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

8. Final simplification 43.2%

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

Alternative 14: 28.5% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 66.7%

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

4. Taylor expanded in im around 0 29.2%

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

Alternative 15: 3.1% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -27 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -27.0)

C

double code(double re, double im) {
	return -27.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -27.0d0
end function

Java

public static double code(double re, double im) {
	return -27.0;
}

Python

def code(re, im):
	return -27.0

Julia

function code(re, im)
	return -27.0
end

MATLAB

function tmp = code(re, im)
	tmp = -27.0;
end

Wolfram

code[re_, im_] := -27.0

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 72.9%

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

   1. +-commutative 72.9%

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

   2. unpow2 72.9%

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

   3. fma-define 72.9%

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

5. Simplified 72.9%

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

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{-28 + \cos re} \]
8. Step-by-step derivation

   1. +-commutative 3.4%

      \[\leadsto \color{blue}{\cos re + -28} \]

9. Simplified 3.4%

   \[\leadsto \color{blue}{\cos re + -28} \]

10. Taylor expanded in re around 0 3.4%

    \[\leadsto \color{blue}{-27} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))