math.sin on complex, imaginary part

Percentage Accurate: 54.1% → 99.7%

Time: 6.3s

Alternatives: 18

Speedup: 1.4×

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

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - 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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.8:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= im_m 3.8)
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (* t_0 (- 1.0 (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im_m <= 3.8) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * (1.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im_m <= 3.8d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    else
        tmp = t_0 * (1.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im_m <= 3.8) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * (1.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im_m <= 3.8:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	else:
		tmp = t_0 * (1.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im_m <= 3.8)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im_m <= 3.8)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	else
		tmp = t_0 * (1.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.8], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7999999999999998

   1. Initial program 33.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 94.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 3.7999999999999998 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im\_m} - e^{im\_m}\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_1 \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_0 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (- (exp (- im_m)) (exp im_m)))
        (t_2 (* t_0 t_1))
        (t_3
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))
   (*
    im_s
    (if (<= t_2 -2e+46)
      (* t_1 0.5)
      (if (<= t_2 0.0)
        (* t_0 t_3)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         t_3))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = exp(-im_m) - exp(im_m);
	double t_2 = t_0 * t_1;
	double t_3 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (t_2 <= -2e+46) {
		tmp = t_1 * 0.5;
	} else if (t_2 <= 0.0) {
		tmp = t_0 * t_3;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * t_3;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m)
	tmp = 0.0
	if (t_2 <= -2e+46)
		tmp = Float64(t_1 * 0.5);
	elseif (t_2 <= 0.0)
		tmp = Float64(t_0 * t_3);
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * t_3);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, -2e+46], N[(t$95$1 * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$0 * t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$3), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2} \]

      4. lower-exp.f64 N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2} \]

      5. sub0-neg N/A

         \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot \frac{1}{2} \]

      6. lift--.f64 N/A

         \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot \frac{1}{2} \]

      7. lift-exp.f64 80.3

         \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot 0.5 \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot \frac{1}{2} \]

      9. sub0-neg N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2} \]

      10. lower-neg.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_2 := \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re)))
        (t_1 (* t_0 (- (exp (- im_m)) (exp im_m))))
        (t_2
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_1 0.0)
        (* t_0 t_2)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         t_2))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im_m) - exp(im_m));
	double t_2 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_1 <= 0.0) {
		tmp = t_0 * t_2;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * t_2;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * t_2);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$0 * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$2), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 91.5%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 91.5

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 77.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 77.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 77.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 77.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 7.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot im\_m\right) \cdot im\_m - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -2e+46)
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_1 0.0)
        (*
         t_0
         (*
          (-
           (*
            (*
             (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333)
             im_m)
            im_m)
           2.0)
          im_m))
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_1 <= -2e+46) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_1 <= 0.0) {
		tmp = t_0 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= -2e+46)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -2e+46], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 90.2

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 99.7

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_0 0.0)
        (* (* (cos re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_0 <= 0.0) {
		tmp = (cos(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(cos(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 90.2

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 99.5

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 99.5%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_0 0.0)
        (* (- (cos re)) im_m)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_0 <= 0.0) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 90.2

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\cos re\right) \cdot im \]

      6. lift-cos.f64 99.4

         \[\leadsto \left(-\cos re\right) \cdot im \]

   5. Applied rewrites 99.4%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666 - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_0 0.0)
        (* (- (* (* im_m im_m) -0.16666666666666666) 1.0) im_m)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_0 <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 90.2

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 99.5

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \frac{-1}{6} - 1\right) \cdot im \]

      5. lift-*.f64 60.8

         \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   8. Applied rewrites 60.8%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. lower-*.f64 68.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\_m\right) \cdot im\_m - 0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666 - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (*
       (fma (* (* re re) 0.020833333333333332) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (- (* (* -0.0003968253968253968 im_m) im_m) 0.016666666666666666)
            (* im_m im_m))
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_0 0.0)
        (* (- (* (* im_m im_m) -0.16666666666666666) 1.0) im_m)
        (*
         (fma (* re re) -0.25 0.5)
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * ((((((((-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_0 <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * im_m) * im_m) - 0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -2e46

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 90.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\cos re}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos re \cdot \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lift-cos.f64 90.2

         \[\leadsto \left(\color{blue}{\cos re} \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   7. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{1}{2}} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, \color{blue}{{re}^{2}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {\color{blue}{re}}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot \left(re \cdot re\right) - \frac{1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot \color{blue}{re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{1}{48}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot im\right) \cdot im - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 6.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 99.5

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \frac{-1}{6} - 1\right) \cdot im \]

      5. lift-*.f64 60.8

         \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   8. Applied rewrites 60.8%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2}, \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot im\right) \cdot im - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))
   (*
    im_s
    (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
      (* 0.5 t_0)
      (* (fma (* re re) -0.25 0.5) t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * t_0;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * t$95$0), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 96.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2}, \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))
   (*
    im_s
    (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
      (* 0.5 t_0)
      (* (* (* re re) -0.25) t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * t_0;
	} else {
		tmp = ((re * re) * -0.25) * t_0;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = 0.5d0 * t_0
    else
        tmp = ((re * re) * (-0.25d0)) * t_0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = 0.5 * t_0;
	} else {
		tmp = ((re * re) * -0.25) * t_0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = 0.5 * t_0
	else:
		tmp = ((re * re) * -0.25) * t_0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(Float64(Float64(re * re) * -0.25) * t_0);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = (((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = 0.5 * t_0;
	else
		tmp = ((re * re) * -0.25) * t_0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * t$95$0), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 96.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2}, \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 26.7

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 26.7%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     0.5
     (*
      (-
       (*
        (-
         (*
          (*
           (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
           im_m)
          im_m)
         0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m))
    (*
     (* (* re re) -0.25)
     (*
      (-
       (*
        (- (* (* -0.016666666666666666 im_m) im_m) 0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = 0.5d0 * (((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    else
        tmp = ((re * re) * (-0.25d0)) * (((((((-0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = 0.5 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	else:
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(Float64(Float64(re * re) * -0.25) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = 0.5 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	else
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 96.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 65.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2}, \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 65.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 26.7

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 26.7%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot im\_m\right) \cdot im\_m - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     0.5
     (*
      (-
       (*
        (* (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333) im_m)
        im_m)
       2.0)
      im_m))
    (*
     (* (* re re) -0.25)
     (*
      (-
       (*
        (- (* (* -0.016666666666666666 im_m) im_m) 0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = 0.5d0 * (((((((-0.016666666666666666d0) * (im_m * im_m)) - 0.3333333333333333d0) * im_m) * im_m) - 2.0d0) * im_m)
    else
        tmp = ((re * re) * (-0.25d0)) * (((((((-0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m)
	else:
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m));
	else
		tmp = Float64(Float64(Float64(re * re) * -0.25) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	else
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 94.2

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \frac{-1}{4} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({re}^{2}, \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. lower-*.f64 67.0

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 65.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   12. Taylor expanded in re around inf

       \[\leadsto \left(\frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. lift-*.f64 26.7

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   14. Applied rewrites 26.7%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot im\_m\right) \cdot im\_m - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -0.001388888888888889\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right) \cdot \left(-im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     0.5
     (*
      (-
       (*
        (* (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333) im_m)
        im_m)
       2.0)
      im_m))
    (*
     (fma
      (- (* (* (* re re) -0.001388888888888889) (* re re)) 0.5)
      (* re re)
      1.0)
     (- im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = fma(((((re * re) * -0.001388888888888889) * (re * re)) - 0.5), (re * re), 1.0) * -im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(re * re) * -0.001388888888888889) * Float64(re * re)) - 0.5), Float64(re * re), 1.0) * Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * (-im$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 94.2

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\cos re\right) \cdot im \]

      6. lift-cos.f64 10.2

         \[\leadsto \left(-\cos re\right) \cdot im \]

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(-\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)\right) \cdot im \]

      2. *-commutative N/A

         \[\leadsto \left(-\left(\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2} + 1\right)\right) \cdot im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      4. lower--.f64 N/A

         \[\leadsto \left(-\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      5. *-commutative N/A

         \[\leadsto \left(-\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      7. +-commutative N/A

         \[\leadsto \left(-\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {re}^{2}, \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      9. unpow2 N/A

         \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      10. lower-*.f64 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      11. unpow2 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot im \]

      13. unpow2 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, re \cdot re, \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, re \cdot re, 1\right)\right) \cdot im \]

      14. lower-*.f64 34.0

          \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right)\right) \cdot im \]

   8. Applied rewrites 34.0%

      \[\leadsto \left(-\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right)\right) \cdot im \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(-\mathsf{fma}\left(\left(\frac{-1}{720} \cdot {re}^{2}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, re \cdot re, 1\right)\right) \cdot im \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(-\mathsf{fma}\left(\left({re}^{2} \cdot \frac{-1}{720}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, re \cdot re, 1\right)\right) \cdot im \]

       2. lower-*.f64 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\left({re}^{2} \cdot \frac{-1}{720}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, re \cdot re, 1\right)\right) \cdot im \]

       3. pow2 N/A

          \[\leadsto \left(-\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot \frac{-1}{720}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, re \cdot re, 1\right)\right) \cdot im \]

       4. lift-*.f64 34.0

          \[\leadsto \left(-\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -0.001388888888888889\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right)\right) \cdot im \]

   11. Applied rewrites 34.0%

       \[\leadsto \left(-\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -0.001388888888888889\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(re \cdot re\right) \cdot -0.001388888888888889\right) \cdot \left(re \cdot re\right) - 0.5, re \cdot re, 1\right) \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot im\_m\right) \cdot im\_m - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     0.5
     (*
      (-
       (*
        (* (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333) im_m)
        im_m)
       2.0)
      im_m))
    (*
     (* (fma -0.5 (* re re) 1.0) (fma (* -0.16666666666666666 im_m) im_m -1.0))
     im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = 0.5 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = (fma(-0.5, (re * re), 1.0) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m));
	else
		tmp = Float64(Float64(fma(-0.5, Float64(re * re), 1.0) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{im}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot im\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot {im}^{2} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. lower-*.f64 94.2

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   5. Applied rewrites 94.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 69.9

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      3. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      4. lower-*.f64 58.9

         \[\leadsto \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   8. Applied rewrites 58.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666 - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (* (- (* (* im_m im_m) -0.16666666666666666) 1.0) im_m)
    (*
     (* (fma -0.5 (* re re) 1.0) (fma (* -0.16666666666666666 im_m) im_m -1.0))
     im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} else {
		tmp = (fma(-0.5, (re * re), 1.0) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 1.0) * im_m);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(re * re), 1.0) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 88.8

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \frac{-1}{6} - 1\right) \cdot im \]

      5. lift-*.f64 58.3

         \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   8. Applied rewrites 58.3%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 69.9

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\left(1 + \frac{-1}{2} \cdot {re}^{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, {re}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      3. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2}, re \cdot re, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      4. lower-*.f64 58.9

         \[\leadsto \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   8. Applied rewrites 58.9%

      \[\leadsto \left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666 - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (* (- (* (* im_m im_m) -0.16666666666666666) 1.0) im_m)
    (* (* (* re re) im_m) 0.5))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} else {
		tmp = ((re * re) * im_m) * 0.5;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = (((im_m * im_m) * (-0.16666666666666666d0)) - 1.0d0) * im_m
    else
        tmp = ((re * re) * im_m) * 0.5d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} else {
		tmp = ((re * re) * im_m) * 0.5;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m
	else:
		tmp = ((re * re) * im_m) * 0.5
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 1.0) * im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * im_m) * 0.5);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	else
		tmp = ((re * re) * im_m) * 0.5;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto \left(\cos re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)\right) \cdot im \]

      9. associate-*r* N/A

         \[\leadsto \left(\cos re \cdot \left(\left(\frac{-1}{6} \cdot im\right) \cdot im + -1\right)\right) \cdot im \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      11. lower-*.f64 88.8

          \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot \frac{-1}{6} - 1\right) \cdot im \]

      5. lift-*.f64 58.3

         \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   8. Applied rewrites 58.3%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\cos re\right) \cdot im \]

      6. lift-cos.f64 10.2

         \[\leadsto \left(-\cos re\right) \cdot im \]

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im + \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{2}, \mathsf{neg}\left(im\right)\right) \]

      9. lower-neg.f64 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot \frac{1}{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot \frac{1}{2} \]

       6. lift-*.f64 20.5

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

   11. Applied rewrites 20.5%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (- im_m)
    (* (* (* re re) im_m) 0.5))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = ((re * re) * im_m) * 0.5;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = -im_m
    else
        tmp = ((re * re) * im_m) * 0.5d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = ((re * re) * im_m) * 0.5;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = -im_m
	else:
		tmp = ((re * re) * im_m) * 0.5
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * im_m) * 0.5);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = -im_m;
	else
		tmp = ((re * re) * im_m) * 0.5;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], (-im$95$m), N[(N[(N[(re * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 37.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\cos re\right) \cdot im \]

      6. lift-cos.f64 68.2

         \[\leadsto \left(-\cos re\right) \cdot im \]

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. lower-neg.f64 42.1

         \[\leadsto -im \]

   8. Applied rewrites 42.1%

      \[\leadsto -im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

      5. lower-neg.f64 N/A

         \[\leadsto \left(-\cos re\right) \cdot im \]

      6. lift-cos.f64 10.2

         \[\leadsto \left(-\cos re\right) \cdot im \]

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im + \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{2}, \mathsf{neg}\left(im\right)\right) \]

      9. lower-neg.f64 27.4

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

   8. Applied rewrites 27.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot \frac{1}{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot \frac{1}{2} \]

       6. lift-*.f64 20.5

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

   11. Applied rewrites 20.5%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.8% accurate, 105.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(-im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (- im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -im_m;
}

Fortran

im\_m =     private
im\_s =     private
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(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * -im_m
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -im_m;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -im_m

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(-im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -im_m;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * (-im$95$m)), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\cos re\right)\right) \cdot im \]

   5. lower-neg.f64 N/A

      \[\leadsto \left(-\cos re\right) \cdot im \]

   6. lift-cos.f64 55.3

      \[\leadsto \left(-\cos re\right) \cdot im \]

5. Applied rewrites 55.3%

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

6. Taylor expanded in re around 0

   \[\leadsto -1 \cdot \color{blue}{im} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. lower-neg.f64 34.7

      \[\leadsto -im \]

8. Applied rewrites 34.7%

   \[\leadsto -im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - 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 (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025037 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))