math.sin on complex, imaginary part

Percentage Accurate: 54.3% → 99.8%

Time: 10.9s

Alternatives: 22

Speedup: 2.6×

• 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

real(8) function code(re, im)
    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: 54.3% 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

real(8) function code(re, im)
    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: 99.8% accurate, 0.5× speedup

Math

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

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 (<= (- (exp (- im_m)) (exp im_m)) -0.5)
    (fma (/ (cos re) (exp im_m)) 0.5 (* (* 0.5 (cos re)) (- (exp im_m))))
    (*
     (fma
      (*
       (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
       (* im_m im_m))
      im_m
      (- im_m))
     (cos re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -0.5) {
		tmp = fma((cos(re) / exp(im_m)), 0.5, ((0.5 * cos(re)) * -exp(im_m)));
	} else {
		tmp = fma((fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666) * (im_m * im_m)), im_m, -im_m) * cos(re);
	}
	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(exp(Float64(-im_m)) - exp(im_m)) <= -0.5)
		tmp = fma(Float64(cos(re) / exp(im_m)), 0.5, Float64(Float64(0.5 * cos(re)) * Float64(-exp(im_m))));
	else
		tmp = Float64(fma(Float64(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666) * Float64(im_m * im_m)), im_m, Float64(-im_m)) * cos(re));
	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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[Cos[re], $MachinePrecision] / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * (-N[Exp[im$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m + (-im$95$m)), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.5

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-exp.f64 N/A

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

      12. lift--.f64 N/A

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

      13. exp-diff N/A

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

      14. exp-0 N/A

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

      15. lift-exp.f64 N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   if -0.5 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 37.3%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.6%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right) \cdot \left(im \cdot im\right), \color{blue}{im}, -im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot im\_m, -0.16666666666666666\right) \cdot im\_m, im\_m, -1\right) \cdot \cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* t_0 (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_1 2e+158)
        (*
         (*
          (fma
           (*
            (fma
             (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
             (* im_m im_m)
             -0.16666666666666666)
            im_m)
           im_m
           -1.0)
          (cos re))
         im_m)
        (* (fma (* re re) -0.25 0.5) t_0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = t_0 * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_1 <= 2e+158) {
		tmp = (fma((fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (im_m * im_m), -0.16666666666666666) * im_m), im_m, -1.0) * cos(re)) * im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(t_0 * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_1 <= 2e+158)
		tmp = Float64(Float64(fma(Float64(fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), Float64(im_m * im_m), -0.16666666666666666) * im_m), im_m, -1.0) * cos(re)) * im_m);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e+158], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999991e158

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 97.7%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]
   9. Step-by-step derivation

   10. Applied rewrites 97.7%

       \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im, im, -1\right)\right) \cdot im \]

   if 1.99999999999999991e158 < (*.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 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-exp.f64 84.8

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

   5. Applied rewrites 84.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im, im, -1\right) \cdot \cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot im\_m, -0.16666666666666666\right) \cdot im\_m, im\_m, -1\right) \cdot \cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 2e+158)
        (*
         (*
          (fma
           (*
            (fma
             (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
             (* im_m im_m)
             -0.16666666666666666)
            im_m)
           im_m
           -1.0)
          (cos re))
         im_m)
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 2e+158) {
		tmp = (fma((fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (im_m * im_m), -0.16666666666666666) * im_m), im_m, -1.0) * cos(re)) * im_m;
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 2e+158)
		tmp = Float64(Float64(fma(Float64(fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), Float64(im_m * im_m), -0.16666666666666666) * im_m), im_m, -1.0) * cos(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+158], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999991e158

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 97.7%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]
   9. Step-by-step derivation

   10. Applied rewrites 97.7%

       \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im, im, -1\right)\right) \cdot im \]

   if 1.99999999999999991e158 < (*.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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 55.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right) \cdot im, im, -1\right) \cdot \cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right), im\_m, -im\_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 2e+158)
        (*
         (fma
          (*
           (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
           (* im_m im_m))
          im_m
          (- im_m))
         (cos re))
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 2e+158) {
		tmp = fma((fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666) * (im_m * im_m)), im_m, -im_m) * cos(re);
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 2e+158)
		tmp = Float64(fma(Float64(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666) * Float64(im_m * im_m)), im_m, Float64(-im_m)) * cos(re));
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+158], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m + (-im$95$m)), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999991e158

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

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

   if 1.99999999999999991e158 < (*.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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 55.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right) \cdot \left(im \cdot im\right), im, -im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot \cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 2e+158)
        (*
         (*
          (fma
           (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
           (* im_m im_m)
           -1.0)
          (cos re))
         im_m)
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 2e+158) {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * cos(re)) * im_m;
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 2e+158)
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * cos(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 2e+158], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.99999999999999991e158

   1. Initial program 10.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 97.5%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

   if 1.99999999999999991e158 < (*.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 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 55.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot \cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 200000.0)
        (* (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m) (cos re))
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 200000.0) {
		tmp = (fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m) * cos(re);
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 200000.0)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m) * cos(re));
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 200000.0], N[(N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2e5

   1. Initial program 8.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 98.7%

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

   if 2e5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 53.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 200000:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 200000.0)
        (* (- (cos re)) im_m)
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 200000.0) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 200000.0)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 200000.0], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 69.1%

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

   if -2e46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2e5

   1. Initial program 8.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 2e5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 53.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 200000:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im\_m \cdot im\_m, -0.008333333333333333\right), im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot 1\right) \cdot im\_m\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im\_m \cdot im\_m\right)}{-im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -0.0005)
      (*
       (*
        (fma
         (fma
          (fma -0.0001984126984126984 (* im_m im_m) -0.008333333333333333)
          (* im_m im_m)
          -0.16666666666666666)
         (* im_m im_m)
         -1.0)
        1.0)
       im_m)
      (if (<= t_0 200000.0)
        (* (- (cos re)) im_m)
        (/
         (*
          (fma
           (fma
            (fma -0.001388888888888889 (* re re) 0.041666666666666664)
            (* re re)
            -0.5)
           (* re re)
           1.0)
          (* im_m im_m))
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (im_m * im_m), -0.008333333333333333), (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * 1.0) * im_m;
	} else if (t_0 <= 200000.0) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im_m * im_m), -0.008333333333333333), Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * 1.0) * im_m);
	elseif (t_0 <= 200000.0)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.0005], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 200000.0], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 85.7%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 64.7%

       \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

   if -5.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2e5

   1. Initial program 8.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 2e5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 53.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot 1\right) \cdot im\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 200000:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot \left(im \cdot im\right)}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m, im\_m \cdot im\_m, im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot \left(im\_m \cdot im\_m\right)}{im\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (fma
       (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
       (* im_m im_m)
       im_m)
      (if (<= t_0 0.0)
        (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m)
        (/ (* (fma (* re re) 0.5 -1.0) (* im_m im_m)) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m), (im_m * im_m), im_m);
	} else if (t_0 <= 0.0) {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m;
	} else {
		tmp = (fma((re * re), 0.5, -1.0) * (im_m * im_m)) / 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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = fma(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m), Float64(im_m * im_m), im_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, -1.0) * Float64(im_m * im_m)) / 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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.3%

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

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

   1. Initial program 6.8%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.2%

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

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

   1. Initial program 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 50.9%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im, im \cdot im, im\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot \left(im \cdot im\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m, im\_m \cdot im\_m, im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e+46)
      (fma
       (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
       (* im_m im_m)
       im_m)
      (if (<= t_0 0.0)
        (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m)
        (* (fma (* re re) 0.5 -1.0) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e+46) {
		tmp = fma((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m), (im_m * im_m), im_m);
	} else if (t_0 <= 0.0) {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m;
	} else {
		tmp = fma((re * re), 0.5, -1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e+46)
		tmp = fma(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m), Float64(im_m * im_m), im_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m);
	else
		tmp = Float64(fma(Float64(re * re), 0.5, -1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+46], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $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))) < -2e46

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.3%

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

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

   1. Initial program 6.8%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 54.2%

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

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

   1. Initial program 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.8%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im, im \cdot im, im\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - e^{im\_m} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot 0.5, \cos re, \left(0.5 \cdot \cos re\right) \cdot \left(-e^{im\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right), im\_m, -im\_m\right) \cdot \cos re\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.5)
      (fma (* t_0 0.5) (cos re) (* (* 0.5 (cos re)) (- (exp im_m))))
      (*
       (fma
        (*
         (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
         (* im_m im_m))
        im_m
        (- im_m))
       (cos re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.5) {
		tmp = fma((t_0 * 0.5), cos(re), ((0.5 * cos(re)) * -exp(im_m)));
	} else {
		tmp = fma((fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666) * (im_m * im_m)), im_m, -im_m) * cos(re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.5)
		tmp = fma(Float64(t_0 * 0.5), cos(re), Float64(Float64(0.5 * cos(re)) * Float64(-exp(im_m))));
	else
		tmp = Float64(fma(Float64(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666) * Float64(im_m * im_m)), im_m, Float64(-im_m)) * cos(re));
	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)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[re], $MachinePrecision] + N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * (-N[Exp[im$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m + (-im$95$m)), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.5

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sub0-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 99.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -0.5 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 37.3%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.6%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right) \cdot \left(im \cdot im\right), \color{blue}{im}, -im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

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

Alternative 12: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right), im\_m, -im\_m\right) \cdot \cos re\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.5)
      (* t_0 (* 0.5 (cos re)))
      (*
       (fma
        (*
         (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
         (* im_m im_m))
        im_m
        (- im_m))
       (cos re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0 * (0.5 * cos(re));
	} else {
		tmp = fma((fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666) * (im_m * im_m)), im_m, -im_m) * cos(re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(t_0 * Float64(0.5 * cos(re)));
	else
		tmp = Float64(fma(Float64(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666) * Float64(im_m * im_m)), im_m, Float64(-im_m)) * cos(re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.5], N[(t$95$0 * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m + (-im$95$m)), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.5

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

      4. lift--.f64 N/A

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

      5. sub0-neg N/A

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

      6. lower-neg.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -0.5 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 37.3%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.6%

      \[\leadsto \cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right) \cdot \left(im \cdot im\right), \color{blue}{im}, -im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

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

Alternative 13: 71.7% accurate, 0.8× speedup

Math

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

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* im_m im_m) -0.008333333333333333)
        (* im_m im_m)
        -0.16666666666666666)
       (* im_m im_m)
       -1.0)
      1.0)
     im_m)
    (/
     (*
      (fma
       (fma
        (fma -0.001388888888888889 (* re re) 0.041666666666666664)
        (* re re)
        -0.5)
       (* re re)
       1.0)
      (* im_m im_m))
     (- 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 (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= 0.0) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (im_m * im_m), -0.008333333333333333), (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * 1.0) * im_m;
	} else {
		tmp = (fma(fma(fma(-0.001388888888888889, (re * re), 0.041666666666666664), (re * re), -0.5), (re * re), 1.0) * (im_m * im_m)) / -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(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= 0.0)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im_m * im_m), -0.008333333333333333), Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * 1.0) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), -0.5), Float64(re * re), 1.0) * Float64(im_m * im_m)) / Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / (-im$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 95.4%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 57.8%

       \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

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

   1. Initial program 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   7. Applied rewrites 54.5%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 52.2%

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

4. Final simplification 56.3%

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

Alternative 14: 71.3% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* im_m im_m) -0.008333333333333333)
        (* im_m im_m)
        -0.16666666666666666)
       (* im_m im_m)
       -1.0)
      1.0)
     im_m)
    (*
     (fma
      (fma (* 0.001388888888888889 (* re re)) (* re re) 0.5)
      (* re re)
      -1.0)
     im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= 0.0) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (im_m * im_m), -0.008333333333333333), (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * 1.0) * im_m;
	} else {
		tmp = fma(fma((0.001388888888888889 * (re * re)), (re * re), 0.5), (re * re), -1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= 0.0)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(im_m * im_m), -0.008333333333333333), Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * 1.0) * im_m);
	else
		tmp = Float64(fma(fma(Float64(0.001388888888888889 * Float64(re * re)), Float64(re * re), 0.5), Float64(re * re), -1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 95.4%

      \[\leadsto \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, im \cdot im, \frac{-1}{120}\right), im \cdot im, \frac{-1}{6}\right), im \cdot im, -1\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 57.8%

       \[\leadsto \left(1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot im \]

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

   1. Initial program 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.3%

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

4. Final simplification 50.2%

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

Alternative 15: 69.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.8%

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

   10. Applied rewrites 55.8%

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

   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 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.3%

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

4. Final simplification 48.7%

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

Alternative 16: 69.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.8%

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

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

   1. Initial program 97.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 10.9

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

   5. Applied rewrites 10.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.3%

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

4. Final simplification 48.7%

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

Alternative 17: 68.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot \left(im\_m \cdot im\_m\right)}{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -0.02)
    (/ (* (fma (* re re) 0.5 -1.0) (* im_m im_m)) im_m)
    (*
     (*
      (fma
       (fma -0.016666666666666666 (* im_m im_m) -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)
     0.5))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = (fma((re * re), 0.5, -1.0) * (im_m * im_m)) / im_m;
	} else {
		tmp = (fma(fma(-0.016666666666666666, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * 0.5;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(fma(Float64(re * re), 0.5, -1.0) * Float64(im_m * im_m)) / im_m);
	else
		tmp = Float64(Float64(fma(fma(-0.016666666666666666, Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * 0.5);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision], N[(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] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 57.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 51.8%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 40.3%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 48.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 76.4%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot \left(im \cdot im\right)}{im}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -0.02)
    (* (fma (* re re) 0.5 -1.0) im_m)
    (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma((re * re), 0.5, -1.0) * im_m;
	} else {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(Float64(re * re), 0.5, -1.0) * im_m);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 57.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.2%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 48.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 71.4%

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

Alternative 19: 61.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{im\_m \cdot im\_m}{im\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) 5e-137)
    (/ (* im_m im_m) im_m)
    (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= 5e-137) {
		tmp = (im_m * im_m) / im_m;
	} else {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= 5e-137)
		tmp = Float64(Float64(im_m * im_m) / im_m);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], 5e-137], N[(N[(im$95$m * im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 5.00000000000000001e-137

   1. Initial program 57.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 1.9%

      \[\leadsto -im \]
   9. Step-by-step derivation

   10. Applied rewrites 31.2%

       \[\leadsto \frac{im \cdot im}{im} \]

   if 5.00000000000000001e-137 < (cos.f64 re)

   1. Initial program 48.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 71.4%

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

Alternative 20: 56.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -5e-310)
    im_m
    (* (fma (* -0.16666666666666666 im_m) im_m -1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -5e-310) {
		tmp = im_m;
	} else {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -5e-310)
		tmp = im_m;
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -5e-310], im$95$m, N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -4.999999999999985e-310

   1. Initial program 57.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 1.9%

      \[\leadsto -im \]
   9. Step-by-step derivation

   10. Applied rewrites 13.8%

       \[\leadsto \color{blue}{im} \]

   if -4.999999999999985e-310 < (cos.f64 re)

   1. Initial program 48.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

   5. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), {im}^{3}, -im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto {im}^{3} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), \color{blue}{{im}^{3}}, -im\right) \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 71.4%

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

Alternative 21: 32.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

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 (<= (cos re) -5e-310) im_m (- 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 (cos(re) <= -5e-310) {
		tmp = im_m;
	} else {
		tmp = -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (cos(re) <= (-5d-310)) then
        tmp = im_m
    else
        tmp = -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 (Math.cos(re) <= -5e-310) {
		tmp = im_m;
	} else {
		tmp = -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 math.cos(re) <= -5e-310:
		tmp = im_m
	else:
		tmp = -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 (cos(re) <= -5e-310)
		tmp = im_m;
	else
		tmp = Float64(-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 (cos(re) <= -5e-310)
		tmp = im_m;
	else
		tmp = -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[Cos[re], $MachinePrecision], -5e-310], im$95$m, (-im$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -4.999999999999985e-310

   1. Initial program 57.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 1.9%

      \[\leadsto -im \]
   9. Step-by-step derivation

   10. Applied rewrites 13.8%

       \[\leadsto \color{blue}{im} \]

   if -4.999999999999985e-310 < (cos.f64 re)

   1. Initial program 48.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 57.5

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

   5. Applied rewrites 57.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.8%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 4.9% accurate, 317.0× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s im_m))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * im_m;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * im_m
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * im_m;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * im_m

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * im_m)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * im_m;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * im$95$m), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cos.f64 55.2

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

5. Applied rewrites 55.2%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.6%

   \[\leadsto -im \]
9. Step-by-step derivation

10. Applied rewrites 5.2%

    \[\leadsto \color{blue}{im} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024331 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))