math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 15

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: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im} + 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= (+ (exp (- im)) (exp im)) 4.0)
     (* t_0 (fma im im 2.0))
     (* 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 * fma(im, im, 2.0);
	} else {
		tmp = t_0 * (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 * fma(im, im, 2.0));
	else
		tmp = Float64(t_0 * Float64(exp(im) + 3.0));
	end
	return 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[(im * im + 2.0), $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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-define 100.0%

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

   5. Simplified 100.0%

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

   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 56.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 4:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% 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 99.7%

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

   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 56.1%

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

4. Final simplification 79.8%

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

Alternative 4: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125000.0)
   (cos re)
   (if (<= im 1.02e+103)
     (* 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 <= 125000.0) {
		tmp = cos(re);
	} else if (im <= 1.02e+103) {
		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 <= 125000.0d0) then
        tmp = cos(re)
    else if (im <= 1.02d+103) 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 <= 125000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.02e+103) {
		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 <= 125000.0:
		tmp = math.cos(re)
	elif im <= 1.02e+103:
		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 <= 125000.0)
		tmp = cos(re);
	elseif (im <= 1.02e+103)
		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 <= 125000.0)
		tmp = cos(re);
	elseif (im <= 1.02e+103)
		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, 125000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.02e+103], 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 < 125000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.7%

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

   if 125000 < im < 1.01999999999999991e103

   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 70.6%

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

   if 1.01999999999999991e103 < 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: 72.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125000.0)
   (cos re)
   (if (<= im 1.02e+103)
     (+ 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 <= 125000.0) {
		tmp = cos(re);
	} else if (im <= 1.02e+103) {
		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 <= 125000.0d0) then
        tmp = cos(re)
    else if (im <= 1.02d+103) 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 <= 125000.0) {
		tmp = Math.cos(re);
	} else if (im <= 1.02e+103) {
		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 <= 125000.0:
		tmp = math.cos(re)
	elif im <= 1.02e+103:
		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 <= 125000.0)
		tmp = cos(re);
	elseif (im <= 1.02e+103)
		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 <= 125000.0)
		tmp = cos(re);
	elseif (im <= 1.02e+103)
		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, 125000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 1.02e+103], 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 < 125000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.7%

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

   if 125000 < im < 1.01999999999999991e103

   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 70.6%

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

      1. distribute-lft-in 70.6%

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

      2. metadata-eval 70.6%

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

   7. Simplified 70.6%

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

   if 1.01999999999999991e103 < 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 6: 71.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125000.0)
   (cos re)
   (if (<= im 4.5e+149)
     (+ 1.5 (* 0.5 (exp im)))
     (* (cos re) (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125000.0) {
		tmp = cos(re);
	} else if (im <= 4.5e+149) {
		tmp = 1.5 + (0.5 * exp(im));
	} else {
		tmp = cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 125000.0d0) then
        tmp = cos(re)
    else if (im <= 4.5d+149) then
        tmp = 1.5d0 + (0.5d0 * exp(im))
    else
        tmp = cos(re) * (2.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 125000.0) {
		tmp = Math.cos(re);
	} else if (im <= 4.5e+149) {
		tmp = 1.5 + (0.5 * Math.exp(im));
	} else {
		tmp = Math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125000.0:
		tmp = math.cos(re)
	elif im <= 4.5e+149:
		tmp = 1.5 + (0.5 * math.exp(im))
	else:
		tmp = math.cos(re) * (2.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 125000.0)
		tmp = cos(re);
	elseif (im <= 4.5e+149)
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	else
		tmp = Float64(cos(re) * Float64(2.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 125000.0)
		tmp = cos(re);
	elseif (im <= 4.5e+149)
		tmp = 1.5 + (0.5 * exp(im));
	else
		tmp = cos(re) * (2.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125000.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.5e+149], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(2.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 125000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.7%

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

   if 125000 < im < 4.49999999999999982e149

   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 69.2%

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

      1. distribute-lft-in 69.2%

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

      2. metadata-eval 69.2%

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

   7. Simplified 69.2%

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

   if 4.49999999999999982e149 < 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 97.6%

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

      1. *-commutative 97.6%

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

      2. +-commutative 97.6%

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

      3. distribute-lft-in 97.6%

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

      4. associate-*r* 97.6%

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

      5. *-commutative 97.6%

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

      6. associate-*r* 97.6%

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

      7. associate-*r* 97.6%

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

      8. distribute-rgt-out 97.6%

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

      9. distribute-lft-out 97.6%

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

      10. *-commutative 97.6%

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

      11. distribute-lft-out 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 125000:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 125000:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 125000.0) (cos re) (+ 1.5 (* 0.5 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 125000.0) {
		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 <= 125000.0d0) 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 <= 125000.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 125000.0:
		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 <= 125000.0)
		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 <= 125000.0)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 125000.0], 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 < 125000

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 72.7%

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

   if 125000 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in re around 0 77.8%

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

      1. distribute-lft-in 77.8%

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

      2. metadata-eval 77.8%

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

   7. Simplified 77.8%

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

Alternative 8: 64.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+89}:\\ \;\;\;\;1 + re \cdot \left(2 - re\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 680.0)
   (cos re)
   (if (<= im 4.8e+89)
     (+ 1.0 (* re (- 2.0 re)))
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = cos(re);
	} else if (im <= 4.8e+89) {
		tmp = 1.0 + (re * (2.0 - re));
	} 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 <= 680.0d0) then
        tmp = cos(re)
    else if (im <= 4.8d+89) then
        tmp = 1.0d0 + (re * (2.0d0 - re))
    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 <= 680.0) {
		tmp = Math.cos(re);
	} else if (im <= 4.8e+89) {
		tmp = 1.0 + (re * (2.0 - re));
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 680.0:
		tmp = math.cos(re)
	elif im <= 4.8e+89:
		tmp = 1.0 + (re * (2.0 - re))
	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 <= 680.0)
		tmp = cos(re);
	elseif (im <= 4.8e+89)
		tmp = Float64(1.0 + Float64(re * Float64(2.0 - re)));
	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 <= 680.0)
		tmp = cos(re);
	elseif (im <= 4.8e+89)
		tmp = 1.0 + (re * (2.0 - re));
	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, 680.0], N[Cos[re], $MachinePrecision], If[LessEqual[im, 4.8e+89], N[(1.0 + N[(re * N[(2.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 680

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.1%

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

   if 680 < im < 4.80000000000000009e89

   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 37.2%

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

      1. *-commutative 37.2%

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

   6. Simplified 37.2%

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

   8. Applied egg-rr 37.2%

      \[\leadsto 1 + \color{blue}{\left(re + \mathsf{fma}\left(-re, re, re\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 37.2%

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

      2. +-commutative 37.2%

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

      3. associate-+r+ 37.2%

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

      4. count-2 37.2%

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

      5. distribute-rgt-out 37.2%

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

      6. unsub-neg 37.2%

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

   10. Simplified 37.2%

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

   if 4.80000000000000009e89 < 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 82.0%

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

      1. distribute-lft-in 82.0%

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

      2. metadata-eval 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in im around 0 74.8%

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

      1. *-commutative 74.8%

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

   10. Simplified 74.8%

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

Alternative 9: 41.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+89}:\\ \;\;\;\;1 + re \cdot \left(2 - re\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480.0)
   1.0
   (if (<= im 8e+89)
     (+ 1.0 (* re (- 2.0 re)))
     (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = 1.0;
	} else if (im <= 8e+89) {
		tmp = 1.0 + (re * (2.0 - re));
	} 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 <= 480.0d0) then
        tmp = 1.0d0
    else if (im <= 8d+89) then
        tmp = 1.0d0 + (re * (2.0d0 - re))
    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 <= 480.0) {
		tmp = 1.0;
	} else if (im <= 8e+89) {
		tmp = 1.0 + (re * (2.0 - re));
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480.0:
		tmp = 1.0
	elif im <= 8e+89:
		tmp = 1.0 + (re * (2.0 - re))
	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 <= 480.0)
		tmp = 1.0;
	elseif (im <= 8e+89)
		tmp = Float64(1.0 + Float64(re * Float64(2.0 - re)));
	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 <= 480.0)
		tmp = 1.0;
	elseif (im <= 8e+89)
		tmp = 1.0 + (re * (2.0 - re));
	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, 480.0], 1.0, If[LessEqual[im, 8e+89], N[(1.0 + N[(re * N[(2.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(im * N[(0.5 + N[(im * N[(0.25 + N[(im * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 480

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.1%

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

   4. Taylor expanded in re around 0 38.8%

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

   if 480 < im < 7.99999999999999996e89

   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 37.2%

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

      1. *-commutative 37.2%

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

   6. Simplified 37.2%

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

   8. Applied egg-rr 37.2%

      \[\leadsto 1 + \color{blue}{\left(re + \mathsf{fma}\left(-re, re, re\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 37.2%

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

      2. +-commutative 37.2%

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

      3. associate-+r+ 37.2%

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

      4. count-2 37.2%

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

      5. distribute-rgt-out 37.2%

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

      6. unsub-neg 37.2%

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

   10. Simplified 37.2%

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

   if 7.99999999999999996e89 < 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 82.0%

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

      1. distribute-lft-in 82.0%

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

      2. metadata-eval 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in im around 0 74.8%

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

      1. *-commutative 74.8%

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

   10. Simplified 74.8%

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

Alternative 10: 39.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 160:\\ \;\;\;\;1\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+152}:\\ \;\;\;\;1 + re \cdot \left(2 - re\right)\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 160.0)
   1.0
   (if (<= im 6.2e+152)
     (+ 1.0 (* re (- 2.0 re)))
     (+ 2.0 (* im (+ 0.5 (* im 0.25)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 160.0) {
		tmp = 1.0;
	} else if (im <= 6.2e+152) {
		tmp = 1.0 + (re * (2.0 - re));
	} 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 <= 160.0d0) then
        tmp = 1.0d0
    else if (im <= 6.2d+152) then
        tmp = 1.0d0 + (re * (2.0d0 - re))
    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 <= 160.0) {
		tmp = 1.0;
	} else if (im <= 6.2e+152) {
		tmp = 1.0 + (re * (2.0 - re));
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 160.0:
		tmp = 1.0
	elif im <= 6.2e+152:
		tmp = 1.0 + (re * (2.0 - re))
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 160.0)
		tmp = 1.0;
	elseif (im <= 6.2e+152)
		tmp = Float64(1.0 + Float64(re * Float64(2.0 - re)));
	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 <= 160.0)
		tmp = 1.0;
	elseif (im <= 6.2e+152)
		tmp = 1.0 + (re * (2.0 - re));
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 160.0], 1.0, If[LessEqual[im, 6.2e+152], N[(1.0 + N[(re * N[(2.0 - re), $MachinePrecision]), $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 < 160

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.1%

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

   4. Taylor expanded in re around 0 38.8%

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

   if 160 < im < 6.2e152

   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 30.2%

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

      1. *-commutative 30.2%

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

   6. Simplified 30.2%

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

   8. Applied egg-rr 30.2%

      \[\leadsto 1 + \color{blue}{\left(re + \mathsf{fma}\left(-re, re, re\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 30.2%

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

      2. +-commutative 30.2%

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

      3. associate-+r+ 30.2%

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

      4. count-2 30.2%

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

      5. distribute-rgt-out 30.2%

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

      6. unsub-neg 30.2%

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

   10. Simplified 30.2%

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

   if 6.2e152 < 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 86.1%

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

      1. distribute-lft-in 86.1%

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

      2. metadata-eval 86.1%

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

   7. Simplified 86.1%

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

   8. Taylor expanded in im around 0 86.1%

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

      1. *-commutative 86.1%

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

   10. Simplified 86.1%

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

Alternative 11: 30.9% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + re \cdot \left(2 - re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 420.0) 1.0 (+ 1.0 (* re (- 2.0 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (re * (2.0 - re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 420.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 + (re * (2.0d0 - re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 + (re * (2.0 - re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = 1.0
	else:
		tmp = 1.0 + (re * (2.0 - re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 + Float64(re * Float64(2.0 - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 420.0)
		tmp = 1.0;
	else
		tmp = 1.0 + (re * (2.0 - re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], 1.0, N[(1.0 + N[(re * N[(2.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 420

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 73.1%

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

   4. Taylor expanded in re around 0 38.8%

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

   if 420 < 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 3.1%

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

   4. Taylor expanded in re around 0 19.3%

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

      1. *-commutative 19.3%

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

   6. Simplified 19.3%

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

   8. Applied egg-rr 19.3%

      \[\leadsto 1 + \color{blue}{\left(re + \mathsf{fma}\left(-re, re, re\right)\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 19.3%

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

      2. +-commutative 19.3%

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

      3. associate-+r+ 19.3%

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

      4. count-2 19.3%

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

      5. distribute-rgt-out 19.3%

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

      6. unsub-neg 19.3%

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

   10. Simplified 19.3%

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

Alternative 12: 28.1% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0 55.6%

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

4. Taylor expanded in re around 0 29.7%

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

Alternative 13: 8.0% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 62.2%

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

5. Applied egg-rr 7.9%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]

7. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{0.25} \]
8. Add Preprocessing

Alternative 14: 7.7% 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 62.2%

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

5. Applied egg-rr 7.9%

   \[\leadsto 0.5 \cdot \color{blue}{0.25} \]

7. Applied egg-rr 7.9%

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

Alternative 15: 3.9% accurate, 308.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0 62.2%

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

5. Applied egg-rr 4.2%

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

   1. metadata-eval 4.2%

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

7. Applied egg-rr 4.2%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))