math.sin on complex, imaginary part

Percentage Accurate: 54.5% → 99.5%

Time: 3.3s

Alternatives: 8

Speedup: 1.2×

• 49.9% 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: 54.5% 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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.3:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-im\_m\right) + 1\right) - e^{im\_m}\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (cos re) 0.5)))
   (*
    im_s
    (if (<= im_m 2.3)
      (* t_0 (* (fma (* im_m im_m) -0.3333333333333333 -2.0) im_m))
      (* (- (+ (- im_m) 1.0) (exp im_m)) 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 = cos(re) * 0.5;
	double tmp;
	if (im_m <= 2.3) {
		tmp = t_0 * (fma((im_m * im_m), -0.3333333333333333, -2.0) * im_m);
	} else {
		tmp = ((-im_m + 1.0) - exp(im_m)) * 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(cos(re) * 0.5)
	tmp = 0.0
	if (im_m <= 2.3)
		tmp = Float64(t_0 * Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * im_m));
	else
		tmp = Float64(Float64(Float64(Float64(-im_m) + 1.0) - exp(im_m)) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2.3], N[(t$95$0 * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-im$95$m) + 1.0), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2999999999999998

   1. Initial program 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 84.3

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

   4. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.3

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 84.3

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

   6. Applied rewrites 84.3%

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

   if 2.2999999999999998 < im

   1. Initial program 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 53.8

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

   4. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 53.8

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 53.8

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

      10. lift-neg.f64 N/A

          \[\leadsto \left(\left(\left(-im\right) + 1\right) - e^{im}\right) \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \left(\left(\left(-im\right) + 1\right) - e^{im}\right) \cdot \mathsf{Rewrite<=}\left(*-commutative, \left(\cos re \cdot \frac{1}{2}\right)\right) \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(\left(\left(-im\right) + 1\right) - e^{im}\right) \cdot \mathsf{Rewrite=>}\left(lower-*.f64, \left(\cos re \cdot \frac{1}{2}\right)\right) \]

   6. Applied rewrites 53.8%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(1 + -1 \cdot im\_m\right) - e^{im\_m}\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \end{array} \end{array} \end{array} \]

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 (- (+ 1.0 (* -1.0 im_m)) (exp im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* 0.5 t_0)
      (if (<= t_1 2e+146)
        (*
         (* (cos re) 0.5)
         (* (fma (* im_m im_m) -0.3333333333333333 -2.0) im_m))
        (* t_0 (fma (* re re) -0.25 0.5)))))))

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 + (-1.0 * im_m)) - exp(im_m);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 2e+146) {
		tmp = (cos(re) * 0.5) * (fma((im_m * im_m), -0.3333333333333333, -2.0) * im_m);
	} else {
		tmp = t_0 * fma((re * re), -0.25, 0.5);
	}
	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(1.0 + Float64(-1.0 * im_m)) - exp(im_m))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 2e+146)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * im_m));
	else
		tmp = Float64(t_0 * fma(Float64(re * re), -0.25, 0.5));
	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[(1.0 + N[(-1.0 * im$95$m), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e+146], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $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 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999987e146

   1. Initial program 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 84.3

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

   4. Applied rewrites 84.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 84.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.3

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 84.3

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

   6. Applied rewrites 84.3%

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

   if 1.99999999999999987e146 < (*.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 54.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. lower-pow.f64 41.1

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 41.1

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      6. lift-neg.f64 41.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 41.1

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 41.1

          \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   6. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.8

         \[\leadsto \left(\left(1 + -1 \cdot \color{blue}{im}\right) - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   9. Applied rewrites 40.8%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+146}:\\ \;\;\;\;-im\_m \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + -1 \cdot im\_m\right) - e^{im\_m}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \end{array} \end{array} \end{array} \]

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 (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -1e-5)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (if (<= t_0 2e+146)
        (- (* im_m (cos re)))
        (* (- (+ 1.0 (* -1.0 im_m)) (exp im_m)) (fma (* re re) -0.25 0.5)))))))

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((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_0 <= -1e-5) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else if (t_0 <= 2e+146) {
		tmp = -(im_m * cos(re));
	} else {
		tmp = ((1.0 + (-1.0 * im_m)) - exp(im_m)) * fma((re * re), -0.25, 0.5);
	}
	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(0.0 - im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -1e-5)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	elseif (t_0 <= 2e+146)
		tmp = Float64(-Float64(im_m * cos(re)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 * im_m)) - exp(im_m)) * fma(Float64(re * re), -0.25, 0.5));
	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[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -1e-5], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+146], (-N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), N[(N[(N[(1.0 + N[(-1.0 * im$95$m), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $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))) < -1.00000000000000008e-5

   1. Initial program 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   if -1.00000000000000008e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999987e146

   1. Initial program 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 51.8

         \[\leadsto -im \cdot \cos re \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   if 1.99999999999999987e146 < (*.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 54.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. lower-pow.f64 41.1

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 41.1

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      6. lift-neg.f64 41.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 41.1

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 41.1

          \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   6. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.8

         \[\leadsto \left(\left(1 + -1 \cdot \color{blue}{im}\right) - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   9. Applied rewrites 40.8%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(1 + -1 \cdot im\_m\right) - e^{im\_m}\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \end{array} \end{array} \end{array} \]

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 (- (+ 1.0 (* -1.0 im_m)) (exp im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* 0.5 t_0)
      (if (<= t_1 0.0)
        (* (fma -0.16666666666666666 (* im_m im_m) -1.0) im_m)
        (* t_0 (fma (* re re) -0.25 0.5)))))))

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 + (-1.0 * im_m)) - exp(im_m);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 0.0) {
		tmp = fma(-0.16666666666666666, (im_m * im_m), -1.0) * im_m;
	} else {
		tmp = t_0 * fma((re * re), -0.25, 0.5);
	}
	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(1.0 + Float64(-1.0 * im_m)) - exp(im_m))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(-0.16666666666666666, Float64(im_m * im_m), -1.0) * im_m);
	else
		tmp = Float64(t_0 * fma(Float64(re * re), -0.25, 0.5));
	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[(1.0 + N[(-1.0 * im$95$m), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $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 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

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

   1. Initial program 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.9

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \]

   7. Applied rewrites 53.9%

      \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{2} - 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.9

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \cdot im \]

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 53.9

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

   9. Applied rewrites 53.9%

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

   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 54.5%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. lower-pow.f64 41.1

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   4. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 41.1

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      5. sub0-neg N/A

         \[\leadsto \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \]

      6. lift-neg.f64 41.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 41.1

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 41.1

          \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   6. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \]

   7. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.8

         \[\leadsto \left(\left(1 + -1 \cdot \color{blue}{im}\right) - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   9. Applied rewrites 40.8%

      \[\leadsto \left(\color{blue}{\left(1 + -1 \cdot im\right)} - e^{im}\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.6% accurate, 0.4× speedup

Math

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

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 (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* 0.5 (- (+ 1.0 (* -1.0 im_m)) (exp im_m)))
      (if (<= t_0 0.0)
        (* (fma -0.16666666666666666 (* im_m im_m) -1.0) im_m)
        (fma (* 0.5 im_m) (* re re) (- 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((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * ((1.0 + (-1.0 * im_m)) - exp(im_m));
	} else if (t_0 <= 0.0) {
		tmp = fma(-0.16666666666666666, (im_m * im_m), -1.0) * im_m;
	} else {
		tmp = fma((0.5 * im_m), (re * re), -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(0.0 - im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(-1.0 * im_m)) - exp(im_m)));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(-0.16666666666666666, Float64(im_m * im_m), -1.0) * im_m);
	else
		tmp = fma(Float64(0.5 * im_m), Float64(re * re), Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

   1. Initial program 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

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

   1. Initial program 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.9

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \]

   7. Applied rewrites 53.9%

      \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{2} - 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.9

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \cdot im \]

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 53.9

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

   9. Applied rewrites 53.9%

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

   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 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1

         \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 36.1

         \[\leadsto \mathsf{fma}\left(0.5 \cdot im, {re}^{2}, -1 \cdot im\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 36.1

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 36.1

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

   9. Applied rewrites 36.1%

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

Alternative 6: 63.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im\_m, re \cdot re, -im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right) \cdot im\_m\\ \end{array} \end{array} \]

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)) -0.002)
    (fma (* 0.5 im_m) (* re re) (- im_m))
    (* (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)) <= -0.002) {
		tmp = fma((0.5 * im_m), (re * re), -im_m);
	} else {
		tmp = 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(0.5 * cos(re)) <= -0.002)
		tmp = fma(Float64(0.5 * im_m), Float64(re * re), Float64(-im_m));
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(0.5 * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im$95$m)), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -2e-3

   1. Initial program 54.5%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.1

         \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]

   7. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{im}, 0.5 \cdot \left(im \cdot {re}^{2}\right)\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 36.1

         \[\leadsto \mathsf{fma}\left(0.5 \cdot im, {re}^{2}, -1 \cdot im\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 36.1

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if -2e-3 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 54.5%

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

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 41.2

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

   4. Applied rewrites 41.2%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 53.9

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \]

   7. Applied rewrites 53.9%

      \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{2} - 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.9

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \cdot im \]

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 53.9

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

   9. Applied rewrites 53.9%

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

Alternative 7: 53.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right) \cdot im\_m\right) \end{array} \]

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 (* (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) {
	return im_s * (fma(-0.16666666666666666, (im_m * im_m), -1.0) * 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(fma(-0.16666666666666666, Float64(im_m * im_m), -1.0) * 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 * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-exp.f64 41.2

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

4. Applied rewrites 41.2%

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

5. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 53.9

      \[\leadsto im \cdot \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \]

7. Applied rewrites 53.9%

   \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{2} - 1\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 53.9

      \[\leadsto \left(-0.16666666666666666 \cdot {im}^{2} - 1\right) \cdot im \]

   4. lift--.f64 N/A

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

   5. sub-flip N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 53.9

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

9. Applied rewrites 53.9%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right) \cdot im \]
10. Add Preprocessing

Alternative 8: 29.6% accurate, 32.7× 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 54.5%

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

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 51.8

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

4. Applied rewrites 51.8%

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

5. Taylor expanded in re around 0

   \[\leadsto -1 \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 29.6%

   \[\leadsto -1 \cdot im \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{im} \]

   2. mul-1-neg N/A

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

   3. lower-neg.f64 29.6

      \[\leadsto -im \]

9. Applied rewrites 29.6%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))