math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 16

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: 76.1% 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 99.7%

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

   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 51.6%

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

4. Final simplification 73.7%

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

Alternative 3: 76.1% 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 99.7%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 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 51.6%

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

4. Final simplification 73.7%

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

Alternative 4: 73.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\\ t_1 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 4.6:\\ \;\;\;\;t\_1 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + t\_0\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(4 + t\_0\right)\\ \end{array} \end{array} \]

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 4.6)
     (*
      t_1
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 t_0)))
     (if (<= im 1.05e+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 <= 4.6) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1.05e+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 <= 4.6d0) then
        tmp = t_1 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + t_0))
    else if (im <= 1.05d+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 <= 4.6) {
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	} else if (im <= 1.05e+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 <= 4.6:
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0))
	elif im <= 1.05e+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 <= 4.6)
		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 <= 1.05e+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 <= 4.6)
		tmp = t_1 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + t_0));
	elseif (im <= 1.05e+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, 4.6], 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, 1.05e+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 < 4.5999999999999996

   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 91.7%

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

   4. Taylor expanded in im around 0 63.0%

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

      1. *-commutative 63.0%

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

   6. Simplified 63.0%

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

   if 4.5999999999999996 < im < 1.0500000000000001e103

   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 70.4%

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

      1. +-commutative 70.4%

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

      2. distribute-lft-in 70.4%

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

      3. metadata-eval 70.4%

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

   7. Simplified 70.4%

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

   if 1.0500000000000001e103 < 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 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 3.1:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 3.1)
     (*
      t_0
      (+
       (+ 1.0 (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))
       (+ 1.0 (* im (+ 1.0 (* 0.5 im))))))
     (if (<= im 1.05e+103)
       (+ (* 0.5 (exp im)) 1.5)
       (*
        t_0
        (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 3.1) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 3.1d0) then
        tmp = t_0 * ((1.0d0 + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))) + (1.0d0 + (im * (1.0d0 + (0.5d0 * im)))))
    else if (im <= 1.05d+103) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = t_0 * (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 t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 3.1) {
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 3.1:
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))))
	elif im <= 1.05e+103:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 3.1)
		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(0.5 * im))))));
	elseif (im <= 1.05e+103)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(t_0 * 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)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 3.1)
		tmp = t_0 * ((1.0 + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))) + (1.0 + (im * (1.0 + (0.5 * im)))));
	elseif (im <= 1.05e+103)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 3.1], 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[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+103], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(t$95$0 * 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 < 3.10000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 91.7%

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

   4. Taylor expanded in im around 0 91.6%

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

   if 3.10000000000000009 < im < 1.0500000000000001e103

   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 70.4%

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

      1. +-commutative 70.4%

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

      2. distribute-lft-in 70.4%

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

      3. metadata-eval 70.4%

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

   7. Simplified 70.4%

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

   if 1.0500000000000001e103 < 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 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right) + \left(1 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;im \leq 4:\\ \;\;\;\;t\_0 \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= im 4.0)
     (* t_0 (+ (+ 1.0 (* im (+ (* 0.5 im) -1.0))) (+ im 1.0)))
     (if (<= im 1.05e+103)
       (+ (* 0.5 (exp im)) 1.5)
       (*
        t_0
        (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im <= 4.0) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * exp(im)) + 1.5;
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im <= 4.0d0) then
        tmp = t_0 * ((1.0d0 + (im * ((0.5d0 * im) + (-1.0d0)))) + (im + 1.0d0))
    else if (im <= 1.05d+103) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    else
        tmp = t_0 * (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 t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im <= 4.0) {
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} else {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im <= 4.0:
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0))
	elif im <= 1.05e+103:
		tmp = (0.5 * math.exp(im)) + 1.5
	else:
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im <= 4.0)
		tmp = Float64(t_0 * Float64(Float64(1.0 + Float64(im * Float64(Float64(0.5 * im) + -1.0))) + Float64(im + 1.0)));
	elseif (im <= 1.05e+103)
		tmp = Float64(Float64(0.5 * exp(im)) + 1.5);
	else
		tmp = Float64(t_0 * 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)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im <= 4.0)
		tmp = t_0 * ((1.0 + (im * ((0.5 * im) + -1.0))) + (im + 1.0));
	elseif (im <= 1.05e+103)
		tmp = (0.5 * exp(im)) + 1.5;
	else
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 4.0], N[(t$95$0 * 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, 1.05e+103], N[(N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision], N[(t$95$0 * 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 < 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 91.7%

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

   4. Taylor expanded in im around 0 91.3%

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

   5. Taylor expanded in im around 0 84.8%

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

   if 4 < im < 1.0500000000000001e103

   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 70.4%

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

      1. +-commutative 70.4%

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

      2. distribute-lft-in 70.4%

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

      3. metadata-eval 70.4%

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

   7. Simplified 70.4%

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

   if 1.0500000000000001e103 < 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 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(1 + im \cdot \left(0.5 \cdot im + -1\right)\right) + \left(im + 1\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.9:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.9)
   (cos re)
   (if (<= im 1.05e+103)
     (+ (* 0.5 (exp im)) 1.5)
     (*
      (* 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 <= 2.9) {
		tmp = cos(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * exp(im)) + 1.5;
	} 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 <= 2.9d0) then
        tmp = cos(re)
    else if (im <= 1.05d+103) then
        tmp = (0.5d0 * exp(im)) + 1.5d0
    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 <= 2.9) {
		tmp = Math.cos(re);
	} else if (im <= 1.05e+103) {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	} 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 <= 2.9:
		tmp = math.cos(re)
	elif im <= 1.05e+103:
		tmp = (0.5 * math.exp(im)) + 1.5
	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 <= 2.9)
		tmp = cos(re);
	elseif (im <= 1.05e+103)
		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(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.9)
		tmp = cos(re);
	elseif (im <= 1.05e+103)
		tmp = (0.5 * exp(im)) + 1.5;
	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, 2.9], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.05e+103], 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[(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 < 2.89999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.7%

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

   if 2.89999999999999991 < im < 1.0500000000000001e103

   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 70.4%

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

      1. +-commutative 70.4%

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

      2. distribute-lft-in 70.4%

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

      3. metadata-eval 70.4%

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

   7. Simplified 70.4%

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

   if 1.0500000000000001e103 < 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. Add Preprocessing

Alternative 8: 72.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.2:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.2)
   (cos re)
   (if (<= im 1.85e+154)
     (+ (* 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 <= 3.2) {
		tmp = cos(re);
	} else if (im <= 1.85e+154) {
		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 <= 3.2d0) then
        tmp = cos(re)
    else if (im <= 1.85d+154) 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 <= 3.2) {
		tmp = Math.cos(re);
	} else if (im <= 1.85e+154) {
		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 <= 3.2:
		tmp = math.cos(re)
	elif im <= 1.85e+154:
		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 <= 3.2)
		tmp = cos(re);
	elseif (im <= 1.85e+154)
		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 <= 3.2)
		tmp = cos(re);
	elseif (im <= 1.85e+154)
		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, 3.2], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.85e+154], 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 < 3.2000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 63.7%

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

   if 3.2000000000000002 < im < 1.84999999999999997e154

   1. Initial program 100.0%

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

   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 76.3%

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

      1. +-commutative 76.3%

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

      2. distribute-lft-in 76.3%

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

      3. metadata-eval 76.3%

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

   7. Simplified 76.3%

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

   if 1.84999999999999997e154 < im

   1. Initial program 100.0%

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

   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: 69.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{im} + 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.0) (cos re) (+ (* 0.5 (exp im)) 1.5)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.0) {
		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 <= 3.0d0) 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 <= 3.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.exp(im)) + 1.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.0:
		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 <= 3.0)
		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 <= 3.0)
		tmp = cos(re);
	else
		tmp = (0.5 * exp(im)) + 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.0], 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

   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 63.7%

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

   if 3 < 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 73.9%

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

      1. +-commutative 73.9%

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

      2. distribute-lft-in 73.9%

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

      3. metadata-eval 73.9%

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

   7. Simplified 73.9%

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

Alternative 10: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 150000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 150000.0)
   (cos re)
   (if (<= im 7.2e+103)
     (+ 1.0 (* (* re re) -0.5))
     (+ (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 150000.0) {
		tmp = cos(re);
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 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 <= 150000.0d0) then
        tmp = cos(re)
    else if (im <= 7.2d+103) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))) + 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 150000.0) {
		tmp = Math.cos(re);
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 150000.0:
		tmp = math.cos(re)
	elif im <= 7.2e+103:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 150000.0)
		tmp = cos(re);
	elseif (im <= 7.2e+103)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))) + 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 150000.0)
		tmp = cos(re);
	elseif (im <= 7.2e+103)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 150000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 7.2e+103], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.5e5

   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 63.7%

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

   if 1.5e5 < im < 7.20000000000000033e103

   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 3.1%

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

   4. Taylor expanded in re around 0 16.8%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 16.8%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot -0.5} \]

   6. Simplified 16.8%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 16.8%

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

   8. Applied egg-rr 16.8%

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

   if 7.20000000000000033e103 < 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 78.0%

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

      1. +-commutative 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in im around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. *-commutative 78.0%

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

   10. Simplified 78.0%

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

Alternative 11: 53.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0185:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0185)
   (*
    0.5
    (+ 2.0 (+ im (* im (+ (* im (+ 0.5 (* im -0.16666666666666666))) -1.0)))))
   (if (<= im 7.2e+103)
     (+ 1.0 (* (* re re) -0.5))
     (+ (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0185) {
		tmp = 0.5 * (2.0 + (im + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0185d0) then
        tmp = 0.5d0 * (2.0d0 + (im + (im * ((im * (0.5d0 + (im * (-0.16666666666666666d0)))) + (-1.0d0)))))
    else if (im <= 7.2d+103) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))) + 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0185) {
		tmp = 0.5 * (2.0 + (im + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0185:
		tmp = 0.5 * (2.0 + (im + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))))
	elif im <= 7.2e+103:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0185)
		tmp = Float64(0.5 * Float64(2.0 + Float64(im + Float64(im * Float64(Float64(im * Float64(0.5 + Float64(im * -0.16666666666666666))) + -1.0)))));
	elseif (im <= 7.2e+103)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))) + 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0185)
		tmp = 0.5 * (2.0 + (im + (im * ((im * (0.5 + (im * -0.16666666666666666))) + -1.0))));
	elseif (im <= 7.2e+103)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0185], N[(0.5 * N[(2.0 + N[(im + N[(im * N[(N[(im * N[(0.5 + N[(im * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.2e+103], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.0184999999999999991

   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 91.8%

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

   4. Taylor expanded in im around 0 91.6%

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

   5. Taylor expanded in re around 0 60.9%

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

   if 0.0184999999999999991 < im < 7.20000000000000033e103

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0 4.3%

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

   4. Taylor expanded in re around 0 16.4%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 16.4%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot -0.5} \]

   6. Simplified 16.4%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 16.4%

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

   8. Applied egg-rr 16.4%

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

   if 7.20000000000000033e103 < 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 78.0%

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

      1. +-commutative 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in im around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. *-commutative 78.0%

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

   10. Simplified 78.0%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0185:\\ \;\;\;\;0.5 \cdot \left(2 + \left(im + im \cdot \left(im \cdot \left(0.5 + im \cdot -0.16666666666666666\right) + -1\right)\right)\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.14e-29)
   1.0
   (if (<= im 7.2e+103)
     (+ 1.0 (* (* re re) -0.5))
     (+ (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.14e-29) {
		tmp = 1.0;
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 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 <= 1.14d-29) then
        tmp = 1.0d0
    else if (im <= 7.2d+103) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    else
        tmp = (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0))))) + 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.14e-29) {
		tmp = 1.0;
	} else if (im <= 7.2e+103) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.14e-29:
		tmp = 1.0
	elif im <= 7.2e+103:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.14e-29)
		tmp = 1.0;
	elseif (im <= 7.2e+103)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	else
		tmp = Float64(Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))) + 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.14e-29)
		tmp = 1.0;
	elseif (im <= 7.2e+103)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333))))) + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.14e-29], 1.0, If[LessEqual[im, 7.2e+103], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.13999999999999995e-29

   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 63.1%

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

   4. Taylor expanded in re around 0 36.4%

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

   if 1.13999999999999995e-29 < im < 7.20000000000000033e103

   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 17.1%

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

   4. Taylor expanded in re around 0 24.6%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 24.6%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot -0.5} \]

   6. Simplified 24.6%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 24.6%

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

   8. Applied egg-rr 24.6%

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

   if 7.20000000000000033e103 < 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 78.0%

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

      1. +-commutative 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in im around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. *-commutative 78.0%

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

   10. Simplified 78.0%

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

Alternative 13: 38.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.14 \cdot 10^{+105}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.14e-29)
   1.0
   (if (<= im 1.14e+105)
     (+ 1.0 (* (* re re) -0.5))
     (+ 2.0 (* im (+ 0.5 (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.14e-29) {
		tmp = 1.0;
	} else if (im <= 1.14e+105) {
		tmp = 1.0 + ((re * re) * -0.5);
	} 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 <= 1.14d-29) then
        tmp = 1.0d0
    else if (im <= 1.14d+105) then
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    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 <= 1.14e-29) {
		tmp = 1.0;
	} else if (im <= 1.14e+105) {
		tmp = 1.0 + ((re * re) * -0.5);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.14e-29:
		tmp = 1.0
	elif im <= 1.14e+105:
		tmp = 1.0 + ((re * re) * -0.5)
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.14e-29)
		tmp = 1.0;
	elseif (im <= 1.14e+105)
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	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 <= 1.14e-29)
		tmp = 1.0;
	elseif (im <= 1.14e+105)
		tmp = 1.0 + ((re * re) * -0.5);
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.14e-29], 1.0, If[LessEqual[im, 1.14e+105], N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $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 < 1.13999999999999995e-29

   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 63.1%

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

   4. Taylor expanded in re around 0 36.4%

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

   if 1.13999999999999995e-29 < im < 1.13999999999999997e105

   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 17.1%

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

   4. Taylor expanded in re around 0 24.6%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 24.6%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot -0.5} \]

   6. Simplified 24.6%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 24.6%

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

   8. Applied egg-rr 24.6%

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

   if 1.13999999999999997e105 < 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 78.0%

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

      1. +-commutative 78.0%

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

      2. distribute-lft-in 78.0%

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

      3. metadata-eval 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in im around 0 55.4%

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

      1. +-commutative 55.4%

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

      2. *-commutative 55.4%

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

   10. Simplified 55.4%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 1.14 \cdot 10^{+105}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.4% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.14 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(re \cdot re\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.14e-29) 1.0 (+ 1.0 (* (* re re) -0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.14e-29) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.14d-29) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + ((re * re) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.14e-29) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + ((re * re) * -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.14e-29:
		tmp = 1.0
	else:
		tmp = 1.0 + ((re * re) * -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.14e-29)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(Float64(re * re) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.14e-29)
		tmp = 1.0;
	else
		tmp = 1.0 + ((re * re) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.14e-29], 1.0, N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.13999999999999995e-29

   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 63.1%

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

   4. Taylor expanded in re around 0 36.4%

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

   if 1.13999999999999995e-29 < 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 9.5%

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

   4. Taylor expanded in re around 0 21.6%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {re}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 21.6%

         \[\leadsto 1 + \color{blue}{{re}^{2} \cdot -0.5} \]

   6. Simplified 21.6%

      \[\leadsto \color{blue}{1 + {re}^{2} \cdot -0.5} \]
   7. Step-by-step derivation

      1. unpow2 21.6%

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

   8. Applied egg-rr 21.6%

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

Alternative 15: 28.4% 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 47.4%

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

4. Taylor expanded in re around 0 27.8%

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

Alternative 16: 9.1% 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 68.8%

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

5. Applied egg-rr 8.9%

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

   1. metadata-eval 8.9%

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

7. Applied egg-rr 8.9%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))