math.sin on complex, imaginary part

Percentage Accurate: 54.6% → 99.9%

Time: 6.6s

Alternatives: 20

Speedup: 1.4×

• 49.8% 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: 54.6% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00115:\\ \;\;\;\;\mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666, \cos re, -\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (<= im_m 0.00115)
    (* (fma (* (* im_m im_m) -0.16666666666666666) (cos re) (- (cos re))) im_m)
    (* (* 0.5 (cos re)) (- (exp (- im_m)) (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 tmp;
	if (im_m <= 0.00115) {
		tmp = fma(((im_m * im_m) * -0.16666666666666666), cos(re), -cos(re)) * im_m;
	} else {
		tmp = (0.5 * cos(re)) * (exp(-im_m) - 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)
	tmp = 0.0
	if (im_m <= 0.00115)
		tmp = Float64(fma(Float64(Float64(im_m * im_m) * -0.16666666666666666), cos(re), Float64(-cos(re))) * im_m);
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(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[im$95$m, 0.00115], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[Cos[re], $MachinePrecision] + (-N[Cos[re], $MachinePrecision])), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.00115

   1. Initial program 38.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 93.6%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   7. 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. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lift-cos.f64 N/A

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

      13. lift-neg.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if 0.00115 < 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. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00115:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.16666666666666666, \cos re, -\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

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

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 (- 1.0 (exp im_m)))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* t_1 (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_2 (- INFINITY))
      (* t_0 0.5)
      (if (<= t_2 0.0)
        (*
         t_1
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -0.0003968253968253968 (* im_m im_m))
                0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m))
        (* (* 0.5 (fma -0.5 (* re re) 1.0)) 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 = 1.0 - exp(im_m);
	double t_1 = 0.5 * cos(re);
	double t_2 = t_1 * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 * 0.5;
	} else if (t_2 <= 0.0) {
		tmp = t_1 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = (0.5 * fma(-0.5, (re * re), 1.0)) * 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(1.0 - exp(im_m))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 * 0.5);
	elseif (t_2 <= 0.0)
		tmp = Float64(t_1 * 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(0.5 * fma(-0.5, Float64(re * re), 1.0)) * 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[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$1 * 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[(0.5 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.0%

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

   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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8%

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

   if -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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 31.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 28.6

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

   8. Applied rewrites 28.6%

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

4. Final simplification 74.3%

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

Alternative 3: 98.8% accurate, 0.4× speedup

Math

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

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 (- 1.0 (exp im_m)))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* t_1 (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_2 (- INFINITY))
      (* t_0 0.5)
      (if (<= t_2 0.0)
        (*
         t_1
         (*
          (-
           (*
            (*
             (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333)
             im_m)
            im_m)
           2.0)
          im_m))
        (* (* 0.5 (fma -0.5 (* re re) 1.0)) 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 = 1.0 - exp(im_m);
	double t_1 = 0.5 * cos(re);
	double t_2 = t_1 * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 * 0.5;
	} else if (t_2 <= 0.0) {
		tmp = t_1 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	} else {
		tmp = (0.5 * fma(-0.5, (re * re), 1.0)) * 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(1.0 - exp(im_m))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(t_1 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 * 0.5);
	elseif (t_2 <= 0.0)
		tmp = Float64(t_1 * 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(0.5 * fma(-0.5, Float64(re * re), 1.0)) * 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[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t$95$1 * 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[(0.5 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.0%

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

   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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 31.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 28.6

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

   8. Applied rewrites 28.6%

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

4. Final simplification 74.3%

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

Alternative 4: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\right) \cdot \left(1 - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (- (exp (- im_m)) (exp im_m))) (t_1 (* (* 0.5 (cos re)) t_0)))
   (*
    im_s
    (if (<= t_1 -1.0)
      (* t_0 0.5)
      (if (<= t_1 0.0)
        (* (* (cos re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (* (* 0.5 (fma -0.5 (* re re) 1.0)) (- 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 = exp(-im_m) - exp(im_m);
	double t_1 = (0.5 * cos(re)) * t_0;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0) {
		tmp = (cos(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = (0.5 * fma(-0.5, (re * re), 1.0)) * (1.0 - 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(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(Float64(0.5 * cos(re)) * t_0)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(cos(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(Float64(0.5 * fma(-0.5, Float64(re * re), 1.0)) * Float64(1.0 - exp(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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -1.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 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[(0.5 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   if -1 < (*.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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 31.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 28.6

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

   8. Applied rewrites 28.6%

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

4. Final simplification 74.2%

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

Alternative 5: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 (- 1.0 (exp im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* t_0 0.5)
      (if (<= t_1 0.0)
        (* (* (cos re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (* (* 0.5 (fma -0.5 (* re re) 1.0)) 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 = 1.0 - exp(im_m);
	double t_1 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0) {
		tmp = (cos(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = (0.5 * fma(-0.5, (re * re), 1.0)) * 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(1.0 - exp(im_m))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(cos(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(Float64(0.5 * fma(-0.5, Float64(re * re), 1.0)) * 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[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, (-Infinity)], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 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[(0.5 * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.0%

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

   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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 31.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 28.6

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

   8. Applied rewrites 28.6%

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

4. Final simplification 74.3%

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

Alternative 6: 98.7% accurate, 0.4× speedup

Math

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

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 (- INFINITY))
      (* (- 1.0 (exp im_m)) 0.5)
      (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 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * 0.5;
	} 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 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * 0.5);
	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, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -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 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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.0%

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

   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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.4%

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

   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 60.2

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

   8. Applied rewrites 60.2%

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

4. Final simplification 83.1%

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

Alternative 7: 98.4% accurate, 0.4× speedup

Math

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

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 (- INFINITY))
      (* (- 1.0 (exp im_m)) 0.5)
      (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 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * 0.5;
	} 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 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * 0.5);
	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, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -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 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 75.8

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

      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 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 76.0%

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

   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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.4%

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

   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 60.2

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

   8. Applied rewrites 60.2%

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

4. Final simplification 83.1%

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

Alternative 8: 95.0% accurate, 0.4× speedup

Math

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

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))))
        (t_1
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -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_0 -1.0)
      (* (fma (* (* re re) 0.020833333333333332) (* re re) 0.5) t_1)
      (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)
         t_1))))))

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 t_1 = (((((((-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_0 <= -1.0) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * t_1;
	} 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) * t_1;
	}
	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)))
	t_1 = 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_0 <= -1.0)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * t_1);
	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) * t_1);
	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]}, Block[{t$95$1 = 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$0, -1.0], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $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] * t$95$1), $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))) < -1

   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 82.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 68.4

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

   11. Applied rewrites 68.4%

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

   if -1 < (*.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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.4%

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

   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 60.2

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

   8. Applied rewrites 60.2%

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

4. Final simplification 81.2%

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

Alternative 9: 74.1% accurate, 0.4× speedup

Math

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

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))))
        (t_1
         (*
          (-
           (*
            (-
             (*
              (*
               (-
                (* -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_0 -1.0)
      (* (fma (* (* re re) 0.020833333333333332) (* re re) 0.5) t_1)
      (if (<= t_0 0.0)
        (- im_m)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         t_1))))))

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 t_1 = (((((((-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_0 <= -1.0) {
		tmp = fma(((re * re) * 0.020833333333333332), (re * re), 0.5) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * t_1;
	}
	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)))
	t_1 = 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_0 <= -1.0)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), Float64(re * re), 0.5) * t_1);
	elseif (t_0 <= 0.0)
		tmp = Float64(-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) * t_1);
	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]}, Block[{t$95$1 = 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$0, -1.0], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-im$95$m), 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$1), $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))) < -1

   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 82.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 68.4

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

   11. Applied rewrites 68.4%

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

   if -1 < (*.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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 5.6

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

      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 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in im 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. lift-neg.f64 50.5

         \[\leadsto -im \]

   8. Applied rewrites 50.5%

      \[\leadsto -im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.4%

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

   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 60.2

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

   8. Applied rewrites 60.2%

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

4. Final simplification 57.5%

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

Alternative 10: 74.1% accurate, 0.4× speedup

Math

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

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 -1.0)
      (*
       (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)
        (*
         (fma
          (-
           (*
            (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
            (* re re))
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (- (* -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 <= -1.0) {
		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;
	} else {
		tmp = fma(((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * (re * re)) - 0.25), (re * re), 0.5) * (((((-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 <= -1.0)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), 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));
	elseif (t_0 <= 0.0)
		tmp = Float64(-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(-0.016666666666666666 * Float64(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, -1.0], 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 * 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], If[LessEqual[t$95$0, 0.0], (-im$95$m), 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[(-0.016666666666666666 * 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]]]), $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))) < -1

   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 82.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 68.4

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

   11. Applied rewrites 68.4%

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

   if -1 < (*.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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 5.6

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

      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 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in im 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. lift-neg.f64 50.5

         \[\leadsto -im \]

   8. Applied rewrites 50.5%

      \[\leadsto -im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in im around 0

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

       1. sub0-neg N/A

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

       2. sinh---cosh-rev N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 62.9

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

   11. Applied rewrites 62.9%

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

   12. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 60.1

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

   14. Applied rewrites 60.1%

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

4. Final simplification 57.5%

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

Alternative 11: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 -1.0)
      (*
       (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)
        (*
         (*
          (fma
           (-
            (*
             (* re re)
             (fma (* -0.001388888888888889 re) re 0.041666666666666664))
            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 t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -1.0) {
		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;
	} else {
		tmp = (fma((((re * re) * fma((-0.001388888888888889 * re), re, 0.041666666666666664)) - 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)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(fma(Float64(Float64(re * re) * 0.020833333333333332), 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));
	elseif (t_0 <= 0.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(re * re) * fma(Float64(-0.001388888888888889 * re), re, 0.041666666666666664)) - 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_] := 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, -1.0], 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 * 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], If[LessEqual[t$95$0, 0.0], (-im$95$m), N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * N[(N[(-0.001388888888888889 * re), $MachinePrecision] * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * 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 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))) < -1

   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 82.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 68.4

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 68.4

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

   11. Applied rewrites 68.4%

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

   if -1 < (*.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 5.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 5.6

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

      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 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in im 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. lift-neg.f64 50.5

         \[\leadsto -im \]

   8. Applied rewrites 50.5%

      \[\leadsto -im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      4. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(re \cdot re\right)\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      6. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, {re}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

   8. Applied rewrites 49.6%

      \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.020833333333333332, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

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)
    (*
     (-
      (*
       (-
        (*
         (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
         (* im_m im_m))
        0.16666666666666666)
       (* im_m im_m))
      1.0)
     im_m)
    (*
     (*
      (fma
       (-
        (*
         (* re re)
         (fma (* -0.001388888888888889 re) re 0.041666666666666664))
        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 = ((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * (im_m * im_m)) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = (fma((((re * re) * fma((-0.001388888888888889 * re), re, 0.041666666666666664)) - 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(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * Float64(im_m * im_m)) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(re * re) * fma(Float64(-0.001388888888888889 * re), re, 0.041666666666666664)) - 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[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * N[(N[(-0.001388888888888889 * re), $MachinePrecision] * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * 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 37.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 29.1

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

      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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im \]

   8. Applied rewrites 55.0%

      \[\leadsto \left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      4. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot \left(re \cdot re\right)\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot {re}^{2} - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      6. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}\right) \cdot {re}^{2} + 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(\frac{-1}{720} \cdot \left(re \cdot re\right) + \frac{1}{24}\right) \cdot \left(re \cdot re\right) - \frac{1}{2}, {re}^{2}, 1\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right)\right) \cdot im \]

   8. Applied rewrites 49.6%

      \[\leadsto \left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot re\right) \cdot \mathsf{fma}\left(-0.001388888888888889 \cdot re, re, 0.041666666666666664\right) - 0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

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)
    (*
     (-
      (*
       (-
        (*
         (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
         (* im_m im_m))
        0.16666666666666666)
       (* im_m im_m))
      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 = ((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * (im_m * im_m)) - 0.16666666666666666) * (im_m * im_m)) - 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(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * Float64(im_m * im_m)) - 0.16666666666666666) * Float64(im_m * im_m)) - 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[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $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 37.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 29.1

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

      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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right) \cdot im \]

   8. Applied rewrites 55.0%

      \[\leadsto \left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   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. pow2 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. lift-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

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) -0.008333333333333333) 0.16666666666666666)
       (* im_m im_m))
      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) * -0.008333333333333333) - 0.16666666666666666) * (im_m * im_m)) - 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(Float64(Float64(im_m * im_m) * -0.008333333333333333) - 0.16666666666666666) * Float64(im_m * im_m)) - 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[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $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 37.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\cos re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot im \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      7. pow2 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

      10. lift-*.f64 52.9

          \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

   8. Applied rewrites 52.9%

      \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   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. pow2 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. lift-*.f64 48.3

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

   8. Applied rewrites 48.3%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, re \cdot re, 1\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

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) -0.008333333333333333) 0.16666666666666666)
       (* im_m im_m))
      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) * -0.008333333333333333) - 0.16666666666666666) * (im_m * im_m)) - 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) * (-0.008333333333333333d0)) - 0.16666666666666666d0) * (im_m * im_m)) - 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) * -0.008333333333333333) - 0.16666666666666666) * (im_m * im_m)) - 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) * -0.008333333333333333) - 0.16666666666666666) * (im_m * im_m)) - 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(Float64(Float64(im_m * im_m) * -0.008333333333333333) - 0.16666666666666666) * Float64(im_m * im_m)) - 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) * -0.008333333333333333) - 0.16666666666666666) * (im_m * im_m)) - 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[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $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 37.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot \color{blue}{im} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot \color{blue}{im} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos re \cdot \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -\cos re\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot im \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \cdot im \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      5. *-commutative N/A

         \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      7. pow2 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot {im}^{2} - 1\right) \cdot im \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{120} - \frac{1}{6}\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

      10. lift-*.f64 52.9

          \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

   8. Applied rewrites 52.9%

      \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 7.6

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

   5. Applied rewrites 7.6%

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

   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. lift-neg.f64 22.6

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

   8. Applied rewrites 22.6%

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

   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 19.4

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

   11. Applied rewrites 19.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.008333333333333333 - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.3% accurate, 0.9× speedup

Math

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

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 37.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 29.1

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

      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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 48.8

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

   8. Applied rewrites 48.8%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot \color{blue}{im} \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 7.6

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

   5. Applied rewrites 7.6%

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

   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. lift-neg.f64 22.6

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

   8. Applied rewrites 22.6%

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

   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 19.4

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

   11. Applied rewrites 19.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
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 37.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 29.1

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

      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 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in im 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. lift-neg.f64 34.8

         \[\leadsto -im \]

   8. Applied rewrites 34.8%

      \[\leadsto -im \]

   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 99.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 7.6

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

   5. Applied rewrites 7.6%

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

   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. lift-neg.f64 22.6

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

   8. Applied rewrites 22.6%

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

   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 19.4

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

   11. Applied rewrites 19.4%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

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

Alternative 18: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \left(re \cdot im\_m\right), 0.5, -im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0.4996:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right) \cdot \left(re \cdot re\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666 - 1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 -0.0005)
      (fma (* re (* re im_m)) 0.5 (- im_m))
      (if (<= t_0 0.4996)
        (*
         (- (* (fma -0.041666666666666664 (* re re) 0.5) (* re re)) 1.0)
         im_m)
        (* (- (* (* im_m im_m) -0.16666666666666666) 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 t_0 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = fma((re * (re * im_m)), 0.5, -im_m);
	} else if (t_0 <= 0.4996) {
		tmp = ((fma(-0.041666666666666664, (re * re), 0.5) * (re * re)) - 1.0) * im_m;
	} else {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 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)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = fma(Float64(re * Float64(re * im_m)), 0.5, Float64(-im_m));
	elseif (t_0 <= 0.4996)
		tmp = Float64(Float64(Float64(fma(-0.041666666666666664, Float64(re * re), 0.5) * Float64(re * re)) - 1.0) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 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_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.0005], N[(N[(re * N[(re * im$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + (-im$95$m)), $MachinePrecision], If[LessEqual[t$95$0, 0.4996], N[(N[(N[(N[(-0.041666666666666664 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -5.0000000000000001e-4

   1. Initial program 52.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 53.3

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

   5. Applied rewrites 53.3%

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

   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. lift-neg.f64 36.5

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

   8. Applied rewrites 36.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 36.6

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

   10. Applied rewrites 36.6%

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

   if -5.0000000000000001e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.499599999999999989

   1. Initial program 58.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in re around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 51.3

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

   8. Applied rewrites 51.3%

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

   if 0.499599999999999989 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 53.3

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

      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 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 78.9

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

   8. Applied rewrites 78.9%

      \[\leadsto \left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.75:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 3.75)
      (*
       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 <= 3.75) {
		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 <= 3.75d0) 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 <= 3.75) {
		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 <= 3.75:
		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 <= 3.75)
		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 <= 3.75)
		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, 3.75], 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 < 3.75

   1. Initial program 38.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 93.6%

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

   if 3.75 < 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. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.75:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.9% 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 54.4%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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 40.5

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

   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 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in im 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. lift-neg.f64 26.7

      \[\leadsto -im \]

8. Applied rewrites 26.7%

   \[\leadsto -im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))))