math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 12

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.7% 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. Final simplification 100.0%

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

Alternative 2: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} + e^{im}\\ \mathbf{if}\;t_0 \leq 2.0000002:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (exp (- im)) (exp im))))
   (if (<= t_0 2.0000002)
     (+ (cos re) (* (cos re) (* 0.5 (* im im))))
     (* 0.5 t_0))))

C

double code(double re, double im) {
	double t_0 = exp(-im) + exp(im);
	double tmp;
	if (t_0 <= 2.0000002) {
		tmp = cos(re) + (cos(re) * (0.5 * (im * im)));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) + exp(im)
    if (t_0 <= 2.0000002d0) then
        tmp = cos(re) + (cos(re) * (0.5d0 * (im * im)))
    else
        tmp = 0.5d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) + Math.exp(im);
	double tmp;
	if (t_0 <= 2.0000002) {
		tmp = Math.cos(re) + (Math.cos(re) * (0.5 * (im * im)));
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) + math.exp(im)
	tmp = 0
	if t_0 <= 2.0000002:
		tmp = math.cos(re) + (math.cos(re) * (0.5 * (im * im)))
	else:
		tmp = 0.5 * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) + exp(im))
	tmp = 0.0
	if (t_0 <= 2.0000002)
		tmp = Float64(cos(re) + Float64(cos(re) * Float64(0.5 * Float64(im * im))));
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) + exp(im);
	tmp = 0.0;
	if (t_0 <= 2.0000002)
		tmp = cos(re) + (cos(re) * (0.5 * (im * im)));
	else
		tmp = 0.5 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0000002], N[(N[Cos[re], $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000020000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.00000020000000012 < (+.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. Taylor expanded in re around 0 81.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} + e^{im} \leq 2.0000002:\\ \;\;\;\;\cos re + \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 3: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+155}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00046 \lor \neg \left(im \leq 0.0006\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4e+155)
   (* im (* 0.5 (* (cos re) im)))
   (if (or (<= im -0.00046) (not (<= im 0.0006)))
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* (cos re) (+ (* 0.5 (* im im)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4e+155) {
		tmp = im * (0.5 * (cos(re) * im));
	} else if ((im <= -0.00046) || !(im <= 0.0006)) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-4d+155)) then
        tmp = im * (0.5d0 * (cos(re) * im))
    else if ((im <= (-0.00046d0)) .or. (.not. (im <= 0.0006d0))) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = cos(re) * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4e+155) {
		tmp = im * (0.5 * (Math.cos(re) * im));
	} else if ((im <= -0.00046) || !(im <= 0.0006)) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.cos(re) * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4e+155:
		tmp = im * (0.5 * (math.cos(re) * im))
	elif (im <= -0.00046) or not (im <= 0.0006):
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.cos(re) * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4e+155)
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	elseif ((im <= -0.00046) || !(im <= 0.0006))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(cos(re) * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4e+155)
		tmp = im * (0.5 * (cos(re) * im));
	elseif ((im <= -0.00046) || ~((im <= 0.0006)))
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = cos(re) * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4e+155], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.00046], N[Not[LessEqual[im, 0.0006]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.00000000000000003e155

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -4.00000000000000003e155 < im < -4.6000000000000001e-4 or 5.99999999999999947e-4 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 82.8%

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

   if -4.6000000000000001e-4 < im < 5.99999999999999947e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow2 100.0%

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

      4. distribute-lft1-in 99.9%

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

      5. +-commutative 99.9%

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

      6. unpow2 99.9%

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

      7. *-commutative 99.9%

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

      8. unpow2 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \cdot 10^{+155}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00046 \lor \neg \left(im \leq 0.0006\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 4: 80.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ t_1 := re \cdot \left(re \cdot \left(-0.5 \cdot \cosh im\right)\right)\\ \mathbf{if}\;im \leq -1.4 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 19500:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im))))
        (t_1 (* re (* re (* -0.5 (cosh im))))))
   (if (<= im -1.4e+141)
     t_0
     (if (<= im -2.8e+22)
       t_1
       (if (<= im 19500.0)
         (+ (cos re) (* im (* 0.5 im)))
         (if (<= im 1.9e+154) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double t_1 = re * (re * (-0.5 * cosh(im)));
	double tmp;
	if (im <= -1.4e+141) {
		tmp = t_0;
	} else if (im <= -2.8e+22) {
		tmp = t_1;
	} else if (im <= 19500.0) {
		tmp = cos(re) + (im * (0.5 * im));
	} else if (im <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    t_1 = re * (re * ((-0.5d0) * cosh(im)))
    if (im <= (-1.4d+141)) then
        tmp = t_0
    else if (im <= (-2.8d+22)) then
        tmp = t_1
    else if (im <= 19500.0d0) then
        tmp = cos(re) + (im * (0.5d0 * im))
    else if (im <= 1.9d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double t_1 = re * (re * (-0.5 * Math.cosh(im)));
	double tmp;
	if (im <= -1.4e+141) {
		tmp = t_0;
	} else if (im <= -2.8e+22) {
		tmp = t_1;
	} else if (im <= 19500.0) {
		tmp = Math.cos(re) + (im * (0.5 * im));
	} else if (im <= 1.9e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	t_1 = re * (re * (-0.5 * math.cosh(im)))
	tmp = 0
	if im <= -1.4e+141:
		tmp = t_0
	elif im <= -2.8e+22:
		tmp = t_1
	elif im <= 19500.0:
		tmp = math.cos(re) + (im * (0.5 * im))
	elif im <= 1.9e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	t_1 = Float64(re * Float64(re * Float64(-0.5 * cosh(im))))
	tmp = 0.0
	if (im <= -1.4e+141)
		tmp = t_0;
	elseif (im <= -2.8e+22)
		tmp = t_1;
	elseif (im <= 19500.0)
		tmp = Float64(cos(re) + Float64(im * Float64(0.5 * im)));
	elseif (im <= 1.9e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	t_1 = re * (re * (-0.5 * cosh(im)));
	tmp = 0.0;
	if (im <= -1.4e+141)
		tmp = t_0;
	elseif (im <= -2.8e+22)
		tmp = t_1;
	elseif (im <= 19500.0)
		tmp = cos(re) + (im * (0.5 * im));
	elseif (im <= 1.9e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(re * N[(-0.5 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.4e+141], t$95$0, If[LessEqual[im, -2.8e+22], t$95$1, If[LessEqual[im, 19500.0], N[(N[Cos[re], $MachinePrecision] + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.39999999999999996e141 or 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in im around inf 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*r* 98.4%

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

      3. associate-*r* 98.4%

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

      4. *-commutative 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   if -1.39999999999999996e141 < im < -2.8e22 or 19500 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 72.0%

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

   4. Simplified 72.0%

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

   5. Taylor expanded in re around inf 24.0%

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

      1. *-commutative 24.0%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) \cdot -0.25} \]

      2. associate-*l* 24.0%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right)} \]

      3. unpow2 24.0%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right) \]

      4. associate-*l* 24.0%

         \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right)\right)} \]

      5. *-commutative 24.0%

         \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)}\right) \]

   7. Simplified 24.0%

      \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 10.0%

         \[\leadsto re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)\right)} \]

      2. expm1-udef 10.0%

         \[\leadsto re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)} - 1\right)} \]

      3. cosh-undef 10.0%

         \[\leadsto re \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)\right)} - 1\right) \]

   9. Applied egg-rr 10.0%

      \[\leadsto re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 10.0%

          \[\leadsto re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)\right)} \]

       2. expm1-log1p 24.0%

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       3. associate-*r* 24.0%

          \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(\left(-0.25 \cdot 2\right) \cdot \cosh im\right)}\right) \]

       4. metadata-eval 24.0%

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

   11. Simplified 24.0%

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

   if -2.8e22 < im < 19500

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.5%

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

   4. Simplified 95.5%

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

      1. fma-udef 95.5%

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

      2. *-commutative 95.5%

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

      3. associate-*r* 95.5%

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

   6. Applied egg-rr 95.5%

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

   7. Taylor expanded in re around 0 95.2%

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

      1. unpow2 95.2%

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

      2. associate-*r* 95.2%

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

      3. *-commutative 95.2%

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

   9. Simplified 95.2%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \cdot 10^{+141}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 \cdot \cosh im\right)\right)\\ \mathbf{elif}\;im \leq 19500:\\ \;\;\;\;\cos re + im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{if}\;im \leq -7.4 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 \cdot \cosh im\right)\right)\\ \mathbf{elif}\;im \leq 1.45:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (cos re) im)))))
   (if (<= im -7.4e+140)
     t_0
     (if (<= im -2.8e+22)
       (* re (* re (* -0.5 (cosh im))))
       (if (<= im 1.45) (cos re) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (cos(re) * im));
	double tmp;
	if (im <= -7.4e+140) {
		tmp = t_0;
	} else if (im <= -2.8e+22) {
		tmp = re * (re * (-0.5 * cosh(im)));
	} else if (im <= 1.45) {
		tmp = cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 * (cos(re) * im))
    if (im <= (-7.4d+140)) then
        tmp = t_0
    else if (im <= (-2.8d+22)) then
        tmp = re * (re * ((-0.5d0) * cosh(im)))
    else if (im <= 1.45d0) then
        tmp = cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * (Math.cos(re) * im));
	double tmp;
	if (im <= -7.4e+140) {
		tmp = t_0;
	} else if (im <= -2.8e+22) {
		tmp = re * (re * (-0.5 * Math.cosh(im)));
	} else if (im <= 1.45) {
		tmp = Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.cos(re) * im))
	tmp = 0
	if im <= -7.4e+140:
		tmp = t_0
	elif im <= -2.8e+22:
		tmp = re * (re * (-0.5 * math.cosh(im)))
	elif im <= 1.45:
		tmp = math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(cos(re) * im)))
	tmp = 0.0
	if (im <= -7.4e+140)
		tmp = t_0;
	elseif (im <= -2.8e+22)
		tmp = Float64(re * Float64(re * Float64(-0.5 * cosh(im))));
	elseif (im <= 1.45)
		tmp = cos(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (cos(re) * im));
	tmp = 0.0;
	if (im <= -7.4e+140)
		tmp = t_0;
	elseif (im <= -2.8e+22)
		tmp = re * (re * (-0.5 * cosh(im)));
	elseif (im <= 1.45)
		tmp = cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.4e+140], t$95$0, If[LessEqual[im, -2.8e+22], N[(re * N[(re * N[(-0.5 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.45], N[Cos[re], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.40000000000000006e140 or 1.44999999999999996 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in im around inf 69.5%

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

      1. unpow2 69.5%

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

      2. associate-*r* 69.5%

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

      3. associate-*r* 69.5%

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

      4. *-commutative 69.5%

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

      5. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if -7.40000000000000006e140 < im < -2.8e22

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 73.1%

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

   4. Simplified 73.1%

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

   5. Taylor expanded in re around inf 26.9%

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

      1. *-commutative 26.9%

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(e^{im} + e^{-im}\right)\right) \cdot -0.25} \]

      2. associate-*l* 26.9%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right)} \]

      3. unpow2 26.9%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right) \]

      4. associate-*l* 26.9%

         \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(\left(e^{im} + e^{-im}\right) \cdot -0.25\right)\right)} \]

      5. *-commutative 26.9%

         \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)}\right) \]

   7. Simplified 26.9%

      \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 11.5%

         \[\leadsto re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)\right)} \]

      2. expm1-udef 11.5%

         \[\leadsto re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(e^{im} + e^{-im}\right)\right)\right)} - 1\right)} \]

      3. cosh-undef 11.5%

         \[\leadsto re \cdot \left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right)\right)} - 1\right) \]

   9. Applied egg-rr 11.5%

      \[\leadsto re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 11.5%

          \[\leadsto re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)\right)} \]

       2. expm1-log1p 26.9%

          \[\leadsto re \cdot \color{blue}{\left(re \cdot \left(-0.25 \cdot \left(2 \cdot \cosh im\right)\right)\right)} \]

       3. associate-*r* 26.9%

          \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(\left(-0.25 \cdot 2\right) \cdot \cosh im\right)}\right) \]

       4. metadata-eval 26.9%

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

   11. Simplified 26.9%

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

   if -2.8e22 < im < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.1%

      \[\leadsto \color{blue}{\cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.4 \cdot 10^{+140}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;re \cdot \left(re \cdot \left(-0.5 \cdot \cosh im\right)\right)\\ \mathbf{elif}\;im \leq 1.45:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.4 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.4) (not (<= im 1.45)))
   (* im (* 0.5 (* (cos re) im)))
   (cos re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.4) || !(im <= 1.45)) {
		tmp = im * (0.5 * (cos(re) * im));
	} else {
		tmp = cos(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 <= (-1.4d0)) .or. (.not. (im <= 1.45d0))) then
        tmp = im * (0.5d0 * (cos(re) * im))
    else
        tmp = cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.4) || !(im <= 1.45)) {
		tmp = im * (0.5 * (Math.cos(re) * im));
	} else {
		tmp = Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.4) or not (im <= 1.45):
		tmp = im * (0.5 * (math.cos(re) * im))
	else:
		tmp = math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.4) || !(im <= 1.45))
		tmp = Float64(im * Float64(0.5 * Float64(cos(re) * im)));
	else
		tmp = cos(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.4) || ~((im <= 1.45)))
		tmp = im * (0.5 * (cos(re) * im));
	else
		tmp = cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.4], N[Not[LessEqual[im, 1.45]], $MachinePrecision]], N[(im * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cos[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.3999999999999999 or 1.44999999999999996 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in im around inf 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. associate-*r* 51.6%

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

      4. *-commutative 51.6%

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

      5. *-commutative 51.6%

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

   7. Simplified 51.6%

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

   if -1.3999999999999999 < im < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.4 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\cos re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re\\ \end{array} \]

Alternative 7: 76.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.cos(re) * ((0.5 * (im * im)) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.5%

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

4. Simplified 78.5%

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

5. Taylor expanded in re around inf 78.5%

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

   1. *-commutative 78.5%

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

   2. associate-*l* 78.5%

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

   3. unpow2 78.5%

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

   4. distribute-lft1-in 78.5%

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

   5. +-commutative 78.5%

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

   6. unpow2 78.5%

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

   7. *-commutative 78.5%

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

   8. unpow2 78.5%

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

7. Simplified 78.5%

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

8. Final simplification 78.5%

   \[\leadsto \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 8: 71.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{if}\;im \leq -0.00046:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 0.00045:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (* im im)) 1.0)))
   (if (<= im -0.00046)
     (* t_0 (+ 1.0 (* -0.5 (* re re))))
     (if (<= im 0.00045) (cos re) t_0))))

C

double code(double re, double im) {
	double t_0 = (0.5 * (im * im)) + 1.0;
	double tmp;
	if (im <= -0.00046) {
		tmp = t_0 * (1.0 + (-0.5 * (re * re)));
	} else if (im <= 0.00045) {
		tmp = cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (im * im)) + 1.0d0
    if (im <= (-0.00046d0)) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) * (re * re)))
    else if (im <= 0.00045d0) then
        tmp = cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * (im * im)) + 1.0;
	double tmp;
	if (im <= -0.00046) {
		tmp = t_0 * (1.0 + (-0.5 * (re * re)));
	} else if (im <= 0.00045) {
		tmp = Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * (im * im)) + 1.0
	tmp = 0
	if im <= -0.00046:
		tmp = t_0 * (1.0 + (-0.5 * (re * re)))
	elif im <= 0.00045:
		tmp = math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * Float64(im * im)) + 1.0)
	tmp = 0.0
	if (im <= -0.00046)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(re * re))));
	elseif (im <= 0.00045)
		tmp = cos(re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * (im * im)) + 1.0;
	tmp = 0.0;
	if (im <= -0.00046)
		tmp = t_0 * (1.0 + (-0.5 * (re * re)));
	elseif (im <= 0.00045)
		tmp = cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[im, -0.00046], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.00045], N[Cos[re], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.6000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 50.5%

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

   4. Simplified 50.5%

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

   5. Taylor expanded in re around inf 50.5%

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

      1. *-commutative 50.5%

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

      2. associate-*l* 50.5%

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

      3. unpow2 50.5%

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

      4. distribute-lft1-in 50.5%

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

      5. +-commutative 50.5%

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

      6. unpow2 50.5%

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

      7. *-commutative 50.5%

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

      8. unpow2 50.5%

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

   7. Simplified 50.5%

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

   8. Taylor expanded in re around 0 43.3%

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

   10. Simplified 43.3%

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

   if -4.6000000000000001e-4 < im < 4.4999999999999999e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 99.4%

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

   if 4.4999999999999999e-4 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 54.1%

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

   4. Simplified 54.1%

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

   5. Taylor expanded in re around 0 47.8%

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

      1. *-commutative 47.8%

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

      2. unpow2 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00046:\\ \;\;\;\;\left(0.5 \cdot \left(im \cdot im\right) + 1\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 0.00045:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]

Alternative 9: 46.8% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.45 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.45) (not (<= im 1.45))) (* 0.5 (* im im)) 1.0))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.45) || !(im <= 1.45)) {
		tmp = 0.5 * (im * im);
	} 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 ((im <= (-1.45d0)) .or. (.not. (im <= 1.45d0))) then
        tmp = 0.5d0 * (im * im)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.45) || !(im <= 1.45)) {
		tmp = 0.5 * (im * im);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.45) or not (im <= 1.45):
		tmp = 0.5 * (im * im)
	else:
		tmp = 1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.45) || !(im <= 1.45))
		tmp = Float64(0.5 * Float64(im * im));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.45) || ~((im <= 1.45)))
		tmp = 0.5 * (im * im);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.45], N[Not[LessEqual[im, 1.45]], $MachinePrecision]], N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if im < -1.44999999999999996 or 1.44999999999999996 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 51.6%

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

   4. Simplified 51.6%

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

      1. fma-udef 51.6%

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

      2. *-commutative 51.6%

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

      3. associate-*r* 51.6%

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in re around 0 43.0%

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

      1. unpow2 43.0%

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

      2. associate-*r* 43.0%

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

      3. *-commutative 43.0%

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

   9. Simplified 43.0%

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

   10. Taylor expanded in im around inf 43.0%

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

       1. unpow2 43.0%

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

   12. Simplified 43.0%

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

   if -1.44999999999999996 < im < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 98.3%

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

   3. Taylor expanded in re around 0 50.1%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 46.9% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0 78.5%

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

4. Simplified 78.5%

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

5. Taylor expanded in re around 0 47.5%

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

   1. *-commutative 47.5%

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

   2. unpow2 47.5%

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

7. Simplified 47.5%

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

8. Final simplification 47.5%

   \[\leadsto 0.5 \cdot \left(im \cdot im\right) + 1 \]

Alternative 11: 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%

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

3. Applied egg-rr 8.7%

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

4. Taylor expanded in re around 0 8.7%

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

5. Final simplification 8.7%

   \[\leadsto 0.25 \]

Alternative 12: 28.0% 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. Taylor expanded in im around 0 56.6%

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

3. Taylor expanded in re around 0 29.4%

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

4. Final simplification 29.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023172 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))