math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.9%

Time: 10.2s

Alternatives: 9

Speedup: 1.1×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(-2\right) \cdot \sinh im\right) \cdot \sin re\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* (* (- 2.0) (sinh im)) (sin re)) 0.5))

C

double code(double re, double im) {
	return ((-2.0 * sinh(im)) * sin(re)) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-2.0d0 * sinh(im)) * sin(re)) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return ((-2.0 * Math.sinh(im)) * Math.sin(re)) * 0.5;
}

Python

def code(re, im):
	return ((-2.0 * math.sinh(im)) * math.sin(re)) * 0.5

Julia

function code(re, im)
	return Float64(Float64(Float64(Float64(-2.0) * sinh(im)) * sin(re)) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = ((-2.0 * sinh(im)) * sin(re)) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[(N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 65.5%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\left(\left(-2\right) \cdot \sinh im\right) \cdot \sin re\right) \cdot 0.5} \]
4. Add Preprocessing

Alternative 2: 78.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\sin \left(-re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3) (* (sin (- re)) im) (* (* 0.5 (sin re)) (- 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = sin(-re) * im;
	} else {
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.3d0) then
        tmp = sin(-re) * im
    else
        tmp = (0.5d0 * sin(re)) * (1.0d0 - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = Math.sin(-re) * im;
	} else {
		tmp = (0.5 * Math.sin(re)) * (1.0 - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.3:
		tmp = math.sin(-re) * im
	else:
		tmp = (0.5 * math.sin(re)) * (1.0 - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = Float64(sin(Float64(-re)) * im);
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(1.0 - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.3)
		tmp = sin(-re) * im;
	else
		tmp = (0.5 * sin(re)) * (1.0 - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[Sin[(-re)], $MachinePrecision] * im), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   1. Initial program 65.5%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.6%

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

   6. Applied rewrites 51.6%

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

   if 1.30000000000000004 < im

   1. Initial program 65.5%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 40.4%

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

Alternative 3: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\sin \left(-re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* 0.5 re) (- 1.0 (exp im)))
     (if (<= t_0 0.01)
       (* (sin (- re)) im)
       (*
        (* re (fma (* re -0.08333333333333333) re 0.5))
        (* (- 2.0) (sinh im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else if (t_0 <= 0.01) {
		tmp = sin(-re) * im;
	} else {
		tmp = (re * fma((re * -0.08333333333333333), re, 0.5)) * (-2.0 * sinh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	elseif (t_0 <= 0.01)
		tmp = Float64(sin(Float64(-re)) * im);
	else
		tmp = Float64(Float64(re * fma(Float64(re * -0.08333333333333333), re, 0.5)) * Float64(Float64(-2.0) * sinh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[Sin[(-re)], $MachinePrecision] * im), $MachinePrecision], N[(N[(re * N[(N[(re * -0.08333333333333333), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 33.3%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0100000000000000002

   1. Initial program 65.5%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.6%

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

   6. Applied rewrites 51.6%

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

   if 0.0100000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2\right) \cdot \sinh im\\ \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- 2.0) (sinh im))))
   (if (<= (* 0.5 (sin re)) -0.01)
     (* (* re (fma (* re -0.08333333333333333) re 0.5)) t_0)
     (* (* 0.5 re) t_0))))

C

double code(double re, double im) {
	double t_0 = -2.0 * sinh(im);
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (re * fma((re * -0.08333333333333333), re, 0.5)) * t_0;
	} else {
		tmp = (0.5 * re) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(-2.0) * sinh(im))
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(re * fma(Float64(re * -0.08333333333333333), re, 0.5)) * t_0);
	else
		tmp = Float64(Float64(0.5 * re) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(re * N[(N[(re * -0.08333333333333333), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.2%

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

Alternative 5: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;re \cdot \mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* re (fma -1.0 im (* 0.16666666666666666 (* im (pow re 2.0)))))
   (* (* 0.5 re) (* (- 2.0) (sinh im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = re * fma(-1.0, im, (0.16666666666666666 * (im * pow(re, 2.0))));
	} else {
		tmp = (0.5 * re) * (-2.0 * sinh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(re * fma(-1.0, im, Float64(0.16666666666666666 * Float64(im * (re ^ 2.0)))));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(-2.0) * sinh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(re * N[(-1.0 * im + N[(0.16666666666666666 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.5%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(-1 \cdot im + \frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)} \]

   7. Applied rewrites 36.2%

      \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(-1, im, 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.2%

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

Alternative 6: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* re (fma (* re -0.08333333333333333) re 0.5)) (* -2.0 im))
   (* (* 0.5 re) (* (- 2.0) (sinh im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (re * fma((re * -0.08333333333333333), re, 0.5)) * (-2.0 * im);
	} else {
		tmp = (0.5 * re) * (-2.0 * sinh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(re * fma(Float64(re * -0.08333333333333333), re, 0.5)) * Float64(-2.0 * im));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(-2.0) * sinh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(re * N[(N[(re * -0.08333333333333333), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[((-2.0) * N[Sinh[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]

   7. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot \frac{-1}{12}, re, \frac{1}{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   9. Applied rewrites 36.3%

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.2%

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

Alternative 7: 40.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(-2 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -5e-12)
   (* (* 0.5 re) (- 1.0 (exp im)))
   (* (* re (fma (* re -0.08333333333333333) re 0.5)) (* -2.0 im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -5e-12) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = (re * fma((re * -0.08333333333333333), re, 0.5)) * (-2.0 * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -5e-12)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * -0.08333333333333333), re, 0.5)) * Float64(-2.0 * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-12], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * -0.08333333333333333), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.9999999999999997e-12

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 33.3%

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

   if -4.9999999999999997e-12 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]

   7. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot \frac{-1}{12}, re, \frac{1}{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   9. Applied rewrites 36.3%

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

Alternative 8: 34.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 5 \cdot 10^{-308}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(im \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 5e-308)
   (* (* re (fma (* re -0.08333333333333333) re 0.5)) (* -2.0 im))
   (* -1.0 (* im re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 5e-308) {
		tmp = (re * fma((re * -0.08333333333333333), re, 0.5)) * (-2.0 * im);
	} else {
		tmp = -1.0 * (im * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 5e-308)
		tmp = Float64(Float64(re * fma(Float64(re * -0.08333333333333333), re, 0.5)) * Float64(-2.0 * im));
	else
		tmp = Float64(-1.0 * Float64(im * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 5e-308], N[(N[(re * N[(N[(re * -0.08333333333333333), $MachinePrecision] * re + 0.5), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * im), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(im * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999955e-308

   1. Initial program 65.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{-im} - e^{im}\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \left(\left(-2\right) \cdot \sinh im\right)} \]

   7. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot \frac{-1}{12}, re, \frac{1}{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   9. Applied rewrites 36.3%

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot -0.08333333333333333, re, 0.5\right)\right) \cdot \color{blue}{\left(-2 \cdot im\right)} \]

   if 4.99999999999999955e-308 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.5%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 32.5%

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

Alternative 9: 32.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(re, im):
	return -1.0 * (im * re)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 32.5%

   \[\leadsto -1 \cdot \left(im \cdot \color{blue}{re}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025153 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))