math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 14

Speedup: 1.0×

• 49.6% 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: 74.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(-im) + exp(im)) <= 4.0) {
		tmp = cos(re);
	} else {
		tmp = (0.5 * cos(re)) * (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) :: tmp
    if ((exp(-im) + exp(im)) <= 4.0d0) then
        tmp = cos(re)
    else
        tmp = (0.5d0 * cos(re)) * (exp(im) + 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(-im) + Math.exp(im)) <= 4.0) {
		tmp = Math.cos(re);
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp(im) + 3.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision], 4.0], N[Cos[re], $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * 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 99.8%

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

   4. Taylor expanded in re around inf 99.8%

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

   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 48.9%

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

4. Final simplification 77.3%

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

Alternative 3: 72.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.9e-5)
   (cos re)
   (if (<= im 4.4e+102)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (*
      (* 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 <= 4.9e-5) {
		tmp = cos(re);
	} else if (im <= 4.4e+102) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} 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 <= 4.9d-5) then
        tmp = cos(re)
    else if (im <= 4.4d+102) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    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 <= 4.9e-5) {
		tmp = Math.cos(re);
	} else if (im <= 4.4e+102) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} 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 <= 4.9e-5:
		tmp = math.cos(re)
	elif im <= 4.4e+102:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	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 <= 4.9e-5)
		tmp = cos(re);
	elseif (im <= 4.4e+102)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	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 <= 4.9e-5)
		tmp = cos(re);
	elseif (im <= 4.4e+102)
		tmp = 0.5 * (exp(-im) + exp(im));
	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, 4.9e-5], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.4e+102], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $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 < 4.9e-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 71.6%

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

   4. Taylor expanded in re around inf 71.6%

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

   if 4.9e-5 < im < 4.40000000000000015e102

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 73.2%

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

   if 4.40000000000000015e102 < 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 4: 72.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3)
   (cos re)
   (if (<= im 4.4e+102)
     (+ 1.5 (* 0.5 (exp im)))
     (*
      (* 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 <= 3.3) {
		tmp = cos(re);
	} else if (im <= 4.4e+102) {
		tmp = 1.5 + (0.5 * exp(im));
	} 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 <= 3.3d0) then
        tmp = cos(re)
    else if (im <= 4.4d+102) then
        tmp = 1.5d0 + (0.5d0 * exp(im))
    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 <= 3.3) {
		tmp = Math.cos(re);
	} else if (im <= 4.4e+102) {
		tmp = 1.5 + (0.5 * Math.exp(im));
	} 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 <= 3.3:
		tmp = math.cos(re)
	elif im <= 4.4e+102:
		tmp = 1.5 + (0.5 * math.exp(im))
	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 <= 3.3)
		tmp = cos(re);
	elseif (im <= 4.4e+102)
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	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 <= 3.3)
		tmp = cos(re);
	elseif (im <= 4.4e+102)
		tmp = 1.5 + (0.5 * exp(im));
	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, 3.3], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.4e+102], N[(1.5 + N[(0.5 * N[Exp[im], $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 < 3.2999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around inf 71.6%

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

   if 3.2999999999999998 < im < 4.40000000000000015e102

   1. Initial program 99.9%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in re around 0 68.5%

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

      1. distribute-lft-in 68.5%

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

      2. metadata-eval 68.5%

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

   7. Simplified 68.5%

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

   if 4.40000000000000015e102 < 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 5: 71.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.7)
   (cos re)
   (if (<= im 2.35e+151)
     (+ 1.5 (* 0.5 (exp im)))
     (* (* 0.5 (cos re)) (+ 4.0 (* im (+ 1.0 (* 0.5 im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = cos(re);
	} else if (im <= 2.35e+151) {
		tmp = 1.5 + (0.5 * exp(im));
	} 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 <= 2.7d0) then
        tmp = cos(re)
    else if (im <= 2.35d+151) then
        tmp = 1.5d0 + (0.5d0 * exp(im))
    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 <= 2.7) {
		tmp = Math.cos(re);
	} else if (im <= 2.35e+151) {
		tmp = 1.5 + (0.5 * Math.exp(im));
	} 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 <= 2.7:
		tmp = math.cos(re)
	elif im <= 2.35e+151:
		tmp = 1.5 + (0.5 * math.exp(im))
	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 <= 2.7)
		tmp = cos(re);
	elseif (im <= 2.35e+151)
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	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 <= 2.7)
		tmp = cos(re);
	elseif (im <= 2.35e+151)
		tmp = 1.5 + (0.5 * exp(im));
	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, 2.7], N[Cos[re], $MachinePrecision], If[LessEqual[im, 2.35e+151], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $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 < 2.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around inf 71.6%

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

   if 2.7000000000000002 < im < 2.34999999999999995e151

   1. Initial program 99.9%

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in re around 0 67.8%

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

      1. distribute-lft-in 67.8%

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

      2. metadata-eval 67.8%

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

   7. Simplified 67.8%

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

   if 2.34999999999999995e151 < 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%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3) (cos re) (+ 1.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3) {
		tmp = cos(re);
	} else {
		tmp = 1.5 + (0.5 * exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.3d0) then
        tmp = cos(re)
    else
        tmp = 1.5d0 + (0.5d0 * exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.3) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.3:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.3)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around inf 71.6%

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

   if 2.2999999999999998 < 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 98.7%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. distribute-lft-in 78.3%

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

      2. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

Alternative 7: 62.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.8 \cdot 10^{+57}:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 7.8e+57)
   (cos re)
   (+ 2.0 (* im (+ 0.5 (* im (* im 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.8e+57) {
		tmp = cos(re);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.8d+57) then
        tmp = cos(re)
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * (im * 0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.8e+57) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.8e+57:
		tmp = math.cos(re)
	else:
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.8e+57)
		tmp = cos(re);
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.8e+57)
		tmp = cos(re);
	else
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.8e+57], N[Cos[re], $MachinePrecision], N[(2.0 + N[(im * N[(0.5 + N[(im * N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999937e57

   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 69.3%

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

   4. Taylor expanded in re around inf 69.3%

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

   if 7.79999999999999937e57 < 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 80.9%

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

      1. distribute-lft-in 80.9%

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

      2. metadata-eval 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in im around 0 69.0%

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

   9. Taylor expanded in im around inf 69.0%

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

       1. *-commutative 69.0%

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

   11. Simplified 69.0%

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

Alternative 8: 39.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.25)
   1.0
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.25) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.25d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * (0.25d0 + (im * 0.08333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.25) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.25:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.25)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(0.25 + Float64(im * 0.08333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.25)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.25], 1.0, N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around 0 44.6%

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

   if 1.25 < 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 98.7%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. distribute-lft-in 78.3%

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

      2. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in im around 0 60.7%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.7% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.2) 1.0 (+ 2.0 (* im (+ 0.5 (* im (* im 0.08333333333333333)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.2d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * (im * 0.08333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.2) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.2:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.2)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * Float64(im * 0.08333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.2)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.2], 1.0, N[(2.0 + N[(im * N[(0.5 + N[(im * N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around 0 44.6%

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

   if 1.19999999999999996 < 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 98.7%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. distribute-lft-in 78.3%

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

      2. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in im around 0 60.7%

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

   9. Taylor expanded in im around inf 60.7%

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

       1. *-commutative 60.7%

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

   11. Simplified 60.7%

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

Alternative 10: 37.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3) 1.0 (+ 2.0 (* im (+ 0.5 (* im 0.25))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 + (im * (0.5d0 + (im * 0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = 1.0;
	else
		tmp = Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], 1.0, N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 71.6%

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

   4. Taylor expanded in re around 0 44.6%

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

   if 1.30000000000000004 < 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 98.7%

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

   5. Taylor expanded in re around 0 78.3%

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

      1. distribute-lft-in 78.3%

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

      2. metadata-eval 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in im around 0 50.1%

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

      1. *-commutative 50.1%

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

   10. Simplified 50.1%

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

Alternative 11: 27.8% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 57.2%

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

4. Taylor expanded in re around 0 35.8%

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

Alternative 12: 9.0% 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 68.0%

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

5. Applied egg-rr 10.4%

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

   1. metadata-eval 10.4%

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

7. Applied egg-rr 10.4%

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

Alternative 13: 7.6% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.125)

C

double code(double re, double im) {
	return 0.125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.125d0
end function

Java

public static double code(double re, double im) {
	return 0.125;
}

Python

def code(re, im):
	return 0.125

Julia

function code(re, im)
	return 0.125
end

MATLAB

function tmp = code(re, im)
	tmp = 0.125;
end

Wolfram

code[re_, im_] := 0.125

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 68.0%

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

5. Applied egg-rr 8.7%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]
6. Step-by-step derivation

   1. metadata-eval 8.7%

      \[\leadsto \color{blue}{0.125} \]

7. Applied egg-rr 8.7%

   \[\leadsto \color{blue}{0.125} \]
8. Add Preprocessing

Alternative 14: 3.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -8.0)

C

double code(double re, double im) {
	return -8.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -8.0d0
end function

Java

public static double code(double re, double im) {
	return -8.0;
}

Python

def code(re, im):
	return -8.0

Julia

function code(re, im)
	return -8.0
end

MATLAB

function tmp = code(re, im)
	tmp = -8.0;
end

Wolfram

code[re_, im_] := -8.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 32.0%

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

5. Taylor expanded in im around 0 11.9%

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

7. Applied egg-rr 3.2%

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

   1. *-commutative 3.2%

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

9. Simplified 3.2%

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

10. Taylor expanded in re around 0 3.1%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))