math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.4s

Alternatives: 16

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: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im} + e^{im}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5000:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.9999999933341466:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (+ (exp (- im)) (exp im)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 -5000.0)
     (* t_0 (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
     (if (<= t_2 0.9999999933341466) (cos re) (* 0.5 t_1)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = exp(-im) + exp(im);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else if (t_2 <= 0.9999999933341466) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = exp(-im) + exp(im)
    t_2 = t_0 * t_1
    if (t_2 <= (-5000.0d0)) then
        tmp = t_0 * (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    else if (t_2 <= 0.9999999933341466d0) then
        tmp = cos(re)
    else
        tmp = 0.5d0 * t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = Math.exp(-im) + Math.exp(im);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else if (t_2 <= 0.9999999933341466) {
		tmp = Math.cos(re);
	} else {
		tmp = 0.5 * t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = math.exp(-im) + math.exp(im)
	t_2 = t_0 * t_1
	tmp = 0
	if t_2 <= -5000.0:
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	elif t_2 <= 0.9999999933341466:
		tmp = math.cos(re)
	else:
		tmp = 0.5 * t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im)) + exp(im))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= -5000.0)
		tmp = Float64(t_0 * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	elseif (t_2 <= 0.9999999933341466)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = exp(-im) + exp(im);
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (t_2 <= -5000.0)
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	elseif (t_2 <= 0.9999999933341466)
		tmp = cos(re);
	else
		tmp = 0.5 * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5000.0], N[(t$95$0 * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999933341466], N[Cos[re], $MachinePrecision], N[(0.5 * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5e3

   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 41.5%

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

   5. Taylor expanded in im around 0 25.9%

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

      1. *-commutative 25.9%

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

   7. Simplified 25.9%

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

   if -5e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999333414658

   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%

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

   4. Taylor expanded in re around inf 100.0%

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

   if 0.99999999333414658 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.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

   3. Taylor expanded in re around 0 100.0%

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

Alternative 3: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 -5000.0)
     (* t_0 (+ 4.0 (* im (+ 1.0 (* im (+ 0.5 (* im 0.16666666666666666)))))))
     (if (<= t_1 2.0) (cos re) (+ 1.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else if (t_1 <= 2.0) {
		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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = t_0 * (exp(-im) + exp(im))
    if (t_1 <= (-5000.0d0)) then
        tmp = t_0 * (4.0d0 + (im * (1.0d0 + (im * (0.5d0 + (im * 0.16666666666666666d0))))))
    else if (t_1 <= 2.0d0) 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 t_0 = 0.5 * Math.cos(re);
	double t_1 = t_0 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	} else if (t_1 <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = t_0 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))))
	elif t_1 <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(t_0 * Float64(4.0 + Float64(im * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.16666666666666666)))))));
	elseif (t_1 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = t_0 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_0 * (4.0 + (im * (1.0 + (im * (0.5 + (im * 0.16666666666666666))))));
	elseif (t_1 <= 2.0)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(t$95$0 * N[(4.0 + N[(im * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5e3

   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 41.5%

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

   5. Taylor expanded in im around 0 25.9%

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

      1. *-commutative 25.9%

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

   7. Simplified 25.9%

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

   if -5e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   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 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

Alternative 4: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0 \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 -5000.0)
     (* t_0 (+ 4.0 (* im (+ 1.0 (* 0.5 im)))))
     (if (<= t_1 2.0) (cos re) (+ 1.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (0.5 * im))));
	} else if (t_1 <= 2.0) {
		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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = t_0 * (exp(-im) + exp(im))
    if (t_1 <= (-5000.0d0)) then
        tmp = t_0 * (4.0d0 + (im * (1.0d0 + (0.5d0 * im))))
    else if (t_1 <= 2.0d0) 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 t_0 = 0.5 * Math.cos(re);
	double t_1 = t_0 * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 * (4.0 + (im * (1.0 + (0.5 * im))));
	} else if (t_1 <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = t_0 * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_0 * (4.0 + (im * (1.0 + (0.5 * im))))
	elif t_1 <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(t_0 * Float64(4.0 + Float64(im * Float64(1.0 + Float64(0.5 * im)))));
	elseif (t_1 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = t_0 * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_0 * (4.0 + (im * (1.0 + (0.5 * im))));
	elseif (t_1 <= 2.0)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(t$95$0 * N[(4.0 + N[(im * N[(1.0 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5e3

   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 41.5%

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

   5. Taylor expanded in im around 0 43.3%

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

      1. *-commutative 43.3%

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

   7. Simplified 43.3%

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

   if -5e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   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 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -5000:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;2 - {re}^{2}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1.5 + 0.5 \cdot e^{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (- 2.0 (pow re 2.0))
     (if (<= t_0 2.0) (cos re) (+ 1.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 2.0 - pow(re, 2.0);
	} else if (t_0 <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = 1.5 + (0.5 * exp(im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 - Math.pow(re, 2.0);
	} else if (t_0 <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 1.5 + (0.5 * Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 2.0 - math.pow(re, 2.0)
	elif t_0 <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = 1.5 + (0.5 * math.exp(im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(2.0 - (re ^ 2.0));
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(1.5 + Float64(0.5 * exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 2.0 - (re ^ 2.0);
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = 1.5 + (0.5 * exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(2.0 - N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[re], $MachinePrecision], N[(1.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   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 42.6%

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

   5. Taylor expanded in im around 0 3.1%

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

   6. Taylor expanded in re around 0 43.9%

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

      1. mul-1-neg 43.9%

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

      2. unsub-neg 43.9%

         \[\leadsto \color{blue}{2 - {re}^{2}} \]

   8. Simplified 43.9%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   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 99.2%

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

   4. Taylor expanded in re around inf 99.2%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

Alternative 6: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;2 - {re}^{2}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (- 2.0 (pow re 2.0))
     (if (<= t_0 2.0)
       (cos re)
       (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 2.0 - pow(re, 2.0);
	} else if (t_0 <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 2.0 - Math.pow(re, 2.0);
	} else if (t_0 <= 2.0) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 2.0 - math.pow(re, 2.0)
	elif t_0 <= 2.0:
		tmp = math.cos(re)
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(2.0 - (re ^ 2.0));
	elseif (t_0 <= 2.0)
		tmp = cos(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)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 2.0 - (re ^ 2.0);
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(2.0 - N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[re], $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 (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   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 42.6%

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

   5. Taylor expanded in im around 0 3.1%

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

   6. Taylor expanded in re around 0 43.9%

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

      1. mul-1-neg 43.9%

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

      2. unsub-neg 43.9%

         \[\leadsto \color{blue}{2 - {re}^{2}} \]

   8. Simplified 43.9%

      \[\leadsto \color{blue}{2 - {re}^{2}} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   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 99.2%

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

   4. Taylor expanded in re around inf 99.2%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in im around 0 38.6%

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

4. Final simplification 73.2%

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

Alternative 7: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.1)
     -1.0
     (if (<= t_0 2.0)
       1.0
       (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
    if (t_0 <= (-0.1d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2.0d0) 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 t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -0.1:
		tmp = -1.0
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (0.25 + (im * 0.08333333333333333)))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		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)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		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_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], -1.0, If[LessEqual[t$95$0, 2.0], 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 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.10000000000000001

   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 0.8%

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

   5. Applied egg-rr 13.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 99.7%

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

   4. Taylor expanded in re around 0 77.3%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in im around 0 38.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(0.25 + im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.1)
     -1.0
     (if (<= t_0 2.0)
       1.0
       (+ 2.0 (* im (+ 0.5 (* im (* im 0.08333333333333333)))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
    if (t_0 <= (-0.1d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2.0d0) 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 t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -0.1:
		tmp = -1.0
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		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)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * (im * 0.08333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], -1.0, If[LessEqual[t$95$0, 2.0], 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 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.10000000000000001

   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 0.8%

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

   5. Applied egg-rr 13.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 99.7%

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

   4. Taylor expanded in re around 0 77.3%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in im around 0 38.6%

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

   9. Taylor expanded in im around inf 38.6%

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

       1. *-commutative 38.6%

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

   11. Simplified 38.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.1)
     -1.0
     (if (<= t_0 2.0) 1.0 (+ 2.0 (* im (+ 0.5 (* im 0.25))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
    if (t_0 <= (-0.1d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2.0d0) 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 t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -0.1:
		tmp = -1.0
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = 2.0 + (im * (0.5 + (im * 0.25)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		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)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = 2.0 + (im * (0.5 + (im * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], -1.0, If[LessEqual[t$95$0, 2.0], 1.0, 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 (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.10000000000000001

   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 0.8%

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

   5. Applied egg-rr 13.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 99.7%

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

   4. Taylor expanded in re around 0 77.3%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in im around 0 57.9%

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

      1. *-commutative 57.9%

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

   10. Simplified 57.9%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 + im \cdot \left(0.5 + im \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 11: 62.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) 2.0)
   (cos re)
   (+ 2.0 (* im (+ 0.5 (* im (+ 0.25 (* im 0.08333333333333333))))))))

C

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

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= 2.0:
		tmp = math.cos(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 (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 2.0)
		tmp = cos(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 (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= 2.0)
		tmp = cos(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[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[Cos[re], $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 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 84.0%

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

   4. Taylor expanded in re around inf 84.0%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 51.3%

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

   5. Taylor expanded in re around 0 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in im around 0 38.6%

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

4. Final simplification 68.9%

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

Alternative 12: 30.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.1) -1.0 1.0))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im) + exp(im))) <= (-0.1d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im))) <= -0.1) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -0.1:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.1)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.10000000000000001

   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 0.8%

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

   5. Applied egg-rr 13.2%

      \[\leadsto 0.5 \cdot \color{blue}{-2} \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.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

   3. Taylor expanded in im around 0 58.5%

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

   4. Taylor expanded in re around 0 45.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 11.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-8\\ \mathbf{else}:\\ \;\;\;\;0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -4e-308) -8.0 0.75))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -4e-308) {
		tmp = -8.0;
	} else {
		tmp = 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im) + exp(im))) <= (-4d-308)) then
        tmp = -8.0d0
    else
        tmp = 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im))) <= -4e-308) {
		tmp = -8.0;
	} else {
		tmp = 0.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -4e-308:
		tmp = -8.0
	else:
		tmp = 0.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -4e-308)
		tmp = -8.0;
	else
		tmp = 0.75;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -4e-308)
		tmp = -8.0;
	else
		tmp = 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-308], -8.0, 0.75]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.00000000000000013e-308

   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 29.9%

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

   5. Taylor expanded in im around 0 11.2%

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

   7. Applied egg-rr 10.4%

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

      1. *-commutative 10.4%

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

   9. Simplified 10.4%

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

   10. Taylor expanded in re around 0 9.8%

       \[\leadsto \color{blue}{-8} \]

   if -4.00000000000000013e-308 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.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

   3. Taylor expanded in re around 0 87.2%

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

   5. Applied egg-rr 13.1%

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

      1. metadata-eval 13.1%

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

   7. Applied egg-rr 13.1%

      \[\leadsto \color{blue}{0.75} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 30.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-8\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= (cos re) -1e-310) -8.0 1.0))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -1e-310) {
		tmp = -8.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-1d-310)) then
        tmp = -8.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -1e-310) {
		tmp = -8.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -1e-310:
		tmp = -8.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -1e-310)
		tmp = -8.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -1e-310)
		tmp = -8.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -1e-310], -8.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -9.999999999999969e-311

   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 29.9%

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

   5. Taylor expanded in im around 0 11.2%

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

   7. Applied egg-rr 10.4%

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

      1. *-commutative 10.4%

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

   9. Simplified 10.4%

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

   10. Taylor expanded in re around 0 9.8%

       \[\leadsto \color{blue}{-8} \]

   if -9.999999999999969e-311 < (cos.f64 re)

   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 58.5%

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

   4. Taylor expanded in re around 0 45.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 9.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-8\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= (cos re) -1e-310) -8.0 0.125))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -1e-310) {
		tmp = -8.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-1d-310)) then
        tmp = -8.0d0
    else
        tmp = 0.125d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -1e-310) {
		tmp = -8.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -1e-310:
		tmp = -8.0
	else:
		tmp = 0.125
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -1e-310)
		tmp = -8.0;
	else
		tmp = 0.125;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -1e-310)
		tmp = -8.0;
	else
		tmp = 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -1e-310], -8.0, 0.125]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -9.999999999999969e-311

   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 29.9%

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

   5. Taylor expanded in im around 0 11.2%

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

   7. Applied egg-rr 10.4%

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

      1. *-commutative 10.4%

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

   9. Simplified 10.4%

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

   10. Taylor expanded in re around 0 9.8%

       \[\leadsto \color{blue}{-8} \]

   if -9.999999999999969e-311 < (cos.f64 re)

   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 87.2%

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

   5. Applied egg-rr 10.9%

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

      1. metadata-eval 10.9%

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

   7. Applied egg-rr 10.9%

      \[\leadsto \color{blue}{0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 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))))