math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.9%

Time: 6.2s

Alternatives: 16

Speedup: 1.0×

• 49.4% 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: 54.4% 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, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.058:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 0.058)
      (*
       t_0
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m))
      (* t_0 (- (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 t_0 = 0.5 * cos(re);
	double tmp;
	if (im_m <= 0.058) {
		tmp = t_0 * (fma(fma(fma(-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 * (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)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im_m <= 0.058)
		tmp = Float64(t_0 * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	else
		tmp = Float64(t_0 * 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_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 0.058], N[(t$95$0 * N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 < 0.0580000000000000029

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

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

   4. Applied rewrites 93.0%

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

   if 0.0580000000000000029 < im

   1. Initial program 54.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lower-neg.f64 54.4

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

   3. Applied rewrites 54.4%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0152:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 0.0152)
      (*
       t_0
       (*
        (fma
         (fma -0.016666666666666666 (* im_m im_m) -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m))
      (* t_0 (- (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 t_0 = 0.5 * cos(re);
	double tmp;
	if (im_m <= 0.0152) {
		tmp = t_0 * (fma(fma(-0.016666666666666666, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	} else {
		tmp = t_0 * (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)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im_m <= 0.0152)
		tmp = Float64(t_0 * Float64(fma(fma(-0.016666666666666666, Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	else
		tmp = Float64(t_0 * 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_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 0.0152], N[(t$95$0 * N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 < 0.0152

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   if 0.0152 < im

   1. Initial program 54.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lower-neg.f64 54.4

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

   3. Applied rewrites 54.4%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (- (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -0.1)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (if (<= t_1 5e-5)
        (* t_0 (* (fma -0.3333333333333333 (* im_m im_m) -2.0) im_m))
        (* (* re re) (* (- (expm1 im_m)) -0.25)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else if (t_1 <= 5e-5) {
		tmp = t_0 * (fma(-0.3333333333333333, (im_m * im_m), -2.0) * im_m);
	} else {
		tmp = (re * re) * (-expm1(im_m) * -0.25);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(0.0 - im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	elseif (t_1 <= 5e-5)
		tmp = Float64(t_0 * Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * im_m));
	else
		tmp = Float64(Float64(re * re) * Float64(Float64(-expm1(im_m)) * -0.25));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[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, -0.1], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(t$95$0 * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * -0.25), $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))) < -0.10000000000000001

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.9%

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

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lift-neg.f64 41.9

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

   6. Applied rewrites 41.9%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 54.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.1

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

   4. Applied rewrites 84.1%

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

   if 5.00000000000000024e-5 < (*.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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in re around inf

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

      1. sub-negate-rev N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sub-negate-rev N/A

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

      6. metadata-eval N/A

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

      7. exp-prod N/A

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

      8. sub-to-mult N/A

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

      9. sub0-neg N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 13.5%

      \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(-im\_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \end{array} \]

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 -0.0001)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (if (<= t_0 5e-5)
        (* (- im_m) (cos re))
        (* (* re re) (* (- (expm1 im_m)) -0.25)))))))

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 <= -0.0001) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else if (t_0 <= 5e-5) {
		tmp = -im_m * cos(re);
	} else {
		tmp = (re * re) * (-expm1(im_m) * -0.25);
	}
	return im_s * tmp;
}

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)) * (Math.exp((0.0 - im_m)) - Math.exp(im_m));
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else if (t_0 <= 5e-5) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = (re * re) * (-Math.expm1(im_m) * -0.25);
	}
	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)) * (math.exp((0.0 - im_m)) - math.exp(im_m))
	tmp = 0
	if t_0 <= -0.0001:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	elif t_0 <= 5e-5:
		tmp = -im_m * math.cos(re)
	else:
		tmp = (re * re) * (-math.expm1(im_m) * -0.25)
	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 <= -0.0001)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	elseif (t_0 <= 5e-5)
		tmp = Float64(Float64(-im_m) * cos(re));
	else
		tmp = Float64(Float64(re * re) * Float64(Float64(-expm1(im_m)) * -0.25));
	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, -0.0001], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * -0.25), $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))) < -1.00000000000000005e-4

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.9%

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

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lift-neg.f64 41.9

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

   6. Applied rewrites 41.9%

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

   if -1.00000000000000005e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 54.4%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-cos.f64 52.2

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

   4. Applied rewrites 52.2%

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

   if 5.00000000000000024e-5 < (*.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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in re around inf

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

      1. sub-negate-rev N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sub-negate-rev N/A

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

      6. metadata-eval N/A

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

      7. exp-prod N/A

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

      8. sub-to-mult N/A

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

      9. sub0-neg N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 13.5%

      \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -0.1:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- 0.0 im_m)) (exp im_m))) -0.1)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (*
       t_0
       (*
        (fma
         (fma -0.016666666666666666 (* im_m im_m) -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((t_0 * (exp((0.0 - im_m)) - exp(im_m))) <= -0.1) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = t_0 * (fma(fma(-0.016666666666666666, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(0.0 - im_m)) - exp(im_m))) <= -0.1)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(fma(fma(-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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.9%

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

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lift-neg.f64 41.9

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

   6. Applied rewrites 41.9%

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

   if -0.10000000000000001 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

Alternative 6: 76.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \end{array} \]

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 -0.1)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (if (<= t_0 0.0)
        (*
         (*
          (fma
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m)
         0.5)
        (* (* re re) (* (- (expm1 im_m)) -0.25)))))))

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 <= -0.1) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else if (t_0 <= 0.0) {
		tmp = (fma(fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * 0.5;
	} else {
		tmp = (re * re) * (-expm1(im_m) * -0.25);
	}
	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 <= -0.1)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * 0.5);
	else
		tmp = Float64(Float64(re * re) * Float64(Float64(-expm1(im_m)) * -0.25));
	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, -0.1], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * -0.25), $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))) < -0.10000000000000001

   1. Initial program 54.4%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 41.9%

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

      1. lift--.f64 N/A

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

      2. sub0-neg N/A

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

      3. lift-neg.f64 41.9

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

   6. Applied rewrites 41.9%

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

   if -0.10000000000000001 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.3

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 58.3

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

   9. Applied rewrites 58.3%

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

   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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in re around inf

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

      1. sub-negate-rev N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sub-negate-rev N/A

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

      6. metadata-eval N/A

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

      7. exp-prod N/A

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

      8. sub-to-mult N/A

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

      9. sub0-neg N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 13.5%

      \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -\mathsf{expm1}\left(im\_m\right)\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(t\_0 \cdot -0.25\right)\\ \end{array} \end{array} \end{array} \]

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 (- (expm1 im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -2e+45)
      (* t_0 0.5)
      (if (<= t_1 0.0)
        (*
         (*
          (fma
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m)
         0.5)
        (* (* re re) (* t_0 -0.25)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -expm1(im_m);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_1 <= -2e+45) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0) {
		tmp = (fma(fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * 0.5;
	} else {
		tmp = (re * re) * (t_0 * -0.25);
	}
	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(-expm1(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 <= -2e+45)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(fma(fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * 0.5);
	else
		tmp = Float64(Float64(re * re) * Float64(t_0 * -0.25));
	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[(Exp[im$95$m] - 1), $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, -2e+45], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(t$95$0 * -0.25), $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))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{0.5} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.3

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 58.3

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

   9. Applied rewrites 58.3%

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

   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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in re around inf

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

      1. sub-negate-rev N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sub-negate-rev N/A

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

      6. metadata-eval N/A

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

      7. exp-prod N/A

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

      8. sub-to-mult N/A

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

      9. sub0-neg N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 13.5%

      \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -\mathsf{expm1}\left(im\_m\right)\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(t\_0 \cdot -0.25\right)\\ \end{array} \end{array} \end{array} \]

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 (- (expm1 im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -2e+45)
      (* t_0 0.5)
      (if (<= t_1 0.0)
        (* 0.5 (* (fma (* im_m im_m) -0.3333333333333333 -2.0) im_m))
        (* (* re re) (* t_0 -0.25)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -expm1(im_m);
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m));
	double tmp;
	if (t_1 <= -2e+45) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.0) {
		tmp = 0.5 * (fma((im_m * im_m), -0.3333333333333333, -2.0) * im_m);
	} else {
		tmp = (re * re) * (t_0 * -0.25);
	}
	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(-expm1(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 <= -2e+45)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.0)
		tmp = Float64(0.5 * Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * im_m));
	else
		tmp = Float64(Float64(re * re) * Float64(t_0 * -0.25));
	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[(Exp[im$95$m] - 1), $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, -2e+45], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(0.5 * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(t$95$0 * -0.25), $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))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.5

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

   10. Applied rewrites 53.5%

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

   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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in re around inf

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

      1. sub-negate-rev N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. sub-negate-rev N/A

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

      6. metadata-eval N/A

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

      7. exp-prod N/A

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

      8. sub-to-mult N/A

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

      9. sub0-neg N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 13.5%

      \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, re \cdot re, -0.5\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+45)
      (* (- (expm1 im_m)) 0.5)
      (if (<= t_0 0.0)
        (* 0.5 (* (fma (* im_m im_m) -0.3333333333333333 -2.0) im_m))
        (* (fma 0.25 (* re re) -0.5) 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 <= -2e+45) {
		tmp = -expm1(im_m) * 0.5;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * (fma((im_m * im_m), -0.3333333333333333, -2.0) * im_m);
	} else {
		tmp = fma(0.25, (re * re), -0.5) * 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 <= -2e+45)
		tmp = Float64(Float64(-expm1(im_m)) * 0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * im_m));
	else
		tmp = Float64(fma(0.25, Float64(re * re), -0.5) * 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, -2e+45], N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(re * re), $MachinePrecision] + -0.5), $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))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 53.5

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

   10. Applied rewrites 53.5%

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

   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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 15.4

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

   7. Applied rewrites 15.4%

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

Alternative 10: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, re \cdot re, -0.5\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (- 0.0 im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+45)
      (* (- (expm1 im_m)) 0.5)
      (if (<= t_0 0.0)
        (* 0.5 (* -2.0 im_m))
        (* (fma 0.25 (* re re) -0.5) 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 <= -2e+45) {
		tmp = -expm1(im_m) * 0.5;
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * (-2.0 * im_m);
	} else {
		tmp = fma(0.25, (re * re), -0.5) * 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 <= -2e+45)
		tmp = Float64(Float64(-expm1(im_m)) * 0.5);
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(-2.0 * im_m));
	else
		tmp = Float64(fma(0.25, Float64(re * re), -0.5) * 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, -2e+45], N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(re * re), $MachinePrecision] + -0.5), $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))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 29.0%

       \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]

   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 54.4%

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

   2. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub0-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, e^{0 - im} - e^{im}, \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, \mathsf{neg}\left(\left(e^{im} - e^{0 - im}\right)\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{0 - im}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      10. sub-to-mult N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - e^{\left(1 - \frac{im}{0}\right) \cdot 0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      11. exp-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\left(e^{im} - {\left(e^{1 - \frac{im}{0}}\right)}^{0}\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      12. metadata-eval N/A

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

      13. lower-expm1.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot \frac{-1}{4}, -\mathsf{expm1}\left(im\right), \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \frac{1}{2}\right) \]

   4. Applied rewrites 5.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.25, -\mathsf{expm1}\left(im\right), \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5\right)} \]

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 15.4

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

   7. Applied rewrites 15.4%

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

Alternative 11: 65.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(-\mathsf{expm1}\left(im\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m))) -2e+45)
    (* (- (expm1 im_m)) 0.5)
    (* 0.5 (* -2.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45) {
		tmp = -expm1(im_m) * 0.5;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

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((0.0 - im_m)) - Math.exp(im_m))) <= -2e+45) {
		tmp = -Math.expm1(im_m) * 0.5;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - im_m)) - math.exp(im_m))) <= -2e+45:
		tmp = -math.expm1(im_m) * 0.5
	else:
		tmp = 0.5 * (-2.0 * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = Float64(Float64(-expm1(im_m)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(-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_] := N[(im$95$s * If[LessEqual[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], -2e+45], N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) * 0.5), $MachinePrecision], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 29.0%

       \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot -0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m))) -2e+45)
    (* (* (* im_m im_m) im_m) -0.08333333333333333)
    (* 0.5 (* -2.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45) {
		tmp = ((im_m * im_m) * im_m) * -0.08333333333333333;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp((0.0d0 - im_m)) - exp(im_m))) <= (-2d+45)) then
        tmp = ((im_m * im_m) * im_m) * (-0.08333333333333333d0)
    else
        tmp = 0.5d0 * ((-2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - im_m)) - Math.exp(im_m))) <= -2e+45) {
		tmp = ((im_m * im_m) * im_m) * -0.08333333333333333;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - im_m)) - math.exp(im_m))) <= -2e+45:
		tmp = ((im_m * im_m) * im_m) * -0.08333333333333333
	else:
		tmp = 0.5 * (-2.0 * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = Float64(Float64(Float64(im_m * im_m) * im_m) * -0.08333333333333333);
	else
		tmp = Float64(0.5 * Float64(-2.0 * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = ((im_m * im_m) * im_m) * -0.08333333333333333;
	else
		tmp = 0.5 * (-2.0 * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+45], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * -0.08333333333333333), $MachinePrecision], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(im \cdot \left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) - \frac{1}{2}\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(im \cdot \left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) - \frac{1}{2}\right) \cdot im \]

      3. sub-flip N/A

         \[\leadsto \left(im \cdot \left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot im \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) \cdot im + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot im \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\frac{-1}{12} \cdot im - \frac{1}{4}\right) \cdot im + \frac{-1}{2}\right) \cdot im \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{12} \cdot im - \frac{1}{4}, im, \frac{-1}{2}\right) \cdot im \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{12} \cdot im + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), im, \frac{-1}{2}\right) \cdot im \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{12} \cdot im + \frac{-1}{4}, im, \frac{-1}{2}\right) \cdot im \]

      9. lower-fma.f64 34.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.08333333333333333, im, -0.25\right), im, -0.5\right) \cdot im \]

   7. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.08333333333333333, im, -0.25\right), im, -0.5\right) \cdot \color{blue}{im} \]

   8. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{12} \cdot {im}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {im}^{3} \cdot \frac{-1}{12} \]

      2. lower-*.f64 N/A

         \[\leadsto {im}^{3} \cdot \frac{-1}{12} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{12} \]

      4. pow2 N/A

         \[\leadsto \left({im}^{2} \cdot im\right) \cdot \frac{-1}{12} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({im}^{2} \cdot im\right) \cdot \frac{-1}{12} \]

      6. pow2 N/A

         \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{12} \]

      7. lift-*.f64 29.5

         \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot -0.08333333333333333 \]

   10. Applied rewrites 29.5%

       \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot -0.08333333333333333 \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 29.0%

       \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, im\_m, -0.5\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m))) -2e+45)
    (* (fma -0.25 im_m -0.5) im_m)
    (* 0.5 (* -2.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45) {
		tmp = fma(-0.25, im_m, -0.5) * im_m;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = Float64(fma(-0.25, im_m, -0.5) * im_m);
	else
		tmp = Float64(0.5 * Float64(-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_] := N[(im$95$s * If[LessEqual[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], -2e+45], N[(N[(-0.25 * im$95$m + -0.5), $MachinePrecision] * im$95$m), $MachinePrecision], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 27.8

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

   7. Applied rewrites 27.8%

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

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 29.0%

       \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(-0.25 \cdot im\_m\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- 0.0 im_m)) (exp im_m))) -2e+45)
    (* (* -0.25 im_m) im_m)
    (* 0.5 (* -2.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45) {
		tmp = (-0.25 * im_m) * im_m;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp((0.0d0 - im_m)) - exp(im_m))) <= (-2d+45)) then
        tmp = ((-0.25d0) * im_m) * im_m
    else
        tmp = 0.5d0 * ((-2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - im_m)) - Math.exp(im_m))) <= -2e+45) {
		tmp = (-0.25 * im_m) * im_m;
	} else {
		tmp = 0.5 * (-2.0 * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - im_m)) - math.exp(im_m))) <= -2e+45:
		tmp = (-0.25 * im_m) * im_m
	else:
		tmp = 0.5 * (-2.0 * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = Float64(Float64(-0.25 * im_m) * im_m);
	else
		tmp = Float64(0.5 * Float64(-2.0 * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = (-0.25 * im_m) * im_m;
	else
		tmp = 0.5 * (-2.0 * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+45], N[(N[(-0.25 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 27.8

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

   7. Applied rewrites 27.8%

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

   8. Taylor expanded in im around inf

      \[\leadsto \left(\frac{-1}{4} \cdot im\right) \cdot im \]
   9. Step-by-step derivation

      1. lower-*.f64 23.4

         \[\leadsto \left(-0.25 \cdot im\right) \cdot im \]

   10. Applied rewrites 23.4%

       \[\leadsto \left(-0.25 \cdot im\right) \cdot im \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   3. 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. sub-flip N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-flip N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 29.0%

       \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 27.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im\_m} - e^{im\_m}\right) \leq -2 \cdot 10^{+45}:\\ \;\;\;\;\left(-0.25 \cdot im\_m\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot im\_m\\ \end{array} \end{array} \]

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 (- 0.0 im_m)) (exp im_m))) -2e+45)
    (* (* -0.25 im_m) im_m)
    (* -0.5 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((0.0 - im_m)) - exp(im_m))) <= -2e+45) {
		tmp = (-0.25 * im_m) * im_m;
	} else {
		tmp = -0.5 * im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp((0.0d0 - im_m)) - exp(im_m))) <= (-2d+45)) then
        tmp = ((-0.25d0) * im_m) * im_m
    else
        tmp = (-0.5d0) * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp((0.0 - im_m)) - Math.exp(im_m))) <= -2e+45) {
		tmp = (-0.25 * im_m) * im_m;
	} else {
		tmp = -0.5 * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - im_m)) - math.exp(im_m))) <= -2e+45:
		tmp = (-0.25 * im_m) * im_m
	else:
		tmp = -0.5 * 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(0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = Float64(Float64(-0.25 * im_m) * im_m);
	else
		tmp = Float64(-0.5 * im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp((0.0 - im_m)) - exp(im_m))) <= -2e+45)
		tmp = (-0.25 * im_m) * im_m;
	else
		tmp = -0.5 * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+45], N[(N[(-0.25 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision], N[(-0.5 * 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))) < -1.9999999999999999e45

   1. Initial program 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-flip N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 27.8

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

   7. Applied rewrites 27.8%

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

   8. Taylor expanded in im around inf

      \[\leadsto \left(\frac{-1}{4} \cdot im\right) \cdot im \]
   9. Step-by-step derivation

      1. lower-*.f64 23.4

         \[\leadsto \left(-0.25 \cdot im\right) \cdot im \]

   10. Applied rewrites 23.4%

       \[\leadsto \left(-0.25 \cdot im\right) \cdot im \]

   if -1.9999999999999999e45 < (*.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 54.4%

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

   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. *-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. sub0-neg N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. sub-to-mult N/A

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

      7. exp-prod N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 45.8

         \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 9.7

         \[\leadsto -0.5 \cdot im \]

   7. Applied rewrites 9.7%

      \[\leadsto -0.5 \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 9.7% accurate, 16.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(-0.5 \cdot im\_m\right) \end{array} \]

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 (* -0.5 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 * (-0.5 * 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 * ((-0.5d0) * 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 * (-0.5 * 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 * (-0.5 * 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(-0.5 * 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 * (-0.5 * 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[(-0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

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. *-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. sub0-neg N/A

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

   4. sub-negate-rev N/A

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

   5. lower-neg.f64 N/A

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

   6. sub-to-mult N/A

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

   7. exp-prod N/A

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

   8. metadata-eval N/A

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

   9. lower-expm1.f64 45.8

      \[\leadsto \left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5 \]

4. Applied rewrites 45.8%

   \[\leadsto \color{blue}{\left(-\mathsf{expm1}\left(im\right)\right) \cdot 0.5} \]

5. Taylor expanded in im around 0

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

   1. lower-*.f64 9.7

      \[\leadsto -0.5 \cdot im \]

7. Applied rewrites 9.7%

   \[\leadsto -0.5 \cdot \color{blue}{im} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))