math.sin on complex, imaginary part

Percentage Accurate: 53.5% → 99.9%

Time: 14.9s

Alternatives: 25

Speedup: 0.9×

• 49.0% 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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot -0.016666666666666666, -0.3333333333333333\right), im\_m \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot -2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= t_0 -0.2)
      (* t_0 t_1)
      (*
       t_1
       (fma
        (fma im_m (* im_m -0.016666666666666666) -0.3333333333333333)
        (* im_m (* im_m im_m))
        (* im_m -2.0)))))))

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 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * fma(fma(im_m, (im_m * -0.016666666666666666), -0.3333333333333333), (im_m * (im_m * im_m)), (im_m * -2.0));
	}
	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(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * fma(fma(im_m, Float64(im_m * -0.016666666666666666), -0.3333333333333333), Float64(im_m * Float64(im_m * im_m)), Float64(im_m * -2.0)));
	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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(im$95$m * N[(im$95$m * -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

      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. neg-lowering-neg.f64 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 41.0%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 93.7

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

   5. Simplified 93.7%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 93.8

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

   7. Applied egg-rr 93.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot -0.016666666666666666, -0.3333333333333333\right), im \cdot \left(im \cdot im\right), im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ t_2 := im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ t_3 := im\_m \cdot t\_2\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(\mathsf{fma}\left(t\_3, t\_3, -1\right) \cdot \frac{1}{\mathsf{fma}\left(im\_m, t\_2, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m)))
        (t_1 (* t_0 (* 0.5 (cos re))))
        (t_2
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
           -0.16666666666666666)))
        (t_3 (* im_m t_2)))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* t_0 0.5)
      (if (<= t_1 0.1)
        (* im_m (* (cos re) (* (fma t_3 t_3 -1.0) (/ 1.0 (fma im_m t_2 1.0)))))
        (* (- 1.0 (exp im_m)) (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = t_0 * (0.5 * cos(re));
	double t_2 = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666);
	double t_3 = im_m * t_2;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.1) {
		tmp = im_m * (cos(re) * (fma(t_3, t_3, -1.0) * (1.0 / fma(im_m, t_2, 1.0))));
	} else {
		tmp = (1.0 - exp(im_m)) * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(t_0 * Float64(0.5 * cos(re)))
	t_2 = Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666))
	t_3 = Float64(im_m * t_2)
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(cos(re) * Float64(fma(t_3, t_3, -1.0) * Float64(1.0 / fma(im_m, t_2, 1.0)))));
	else
		tmp = Float64(Float64(1.0 - exp(im_m)) * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[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]}, Block[{t$95$2 = N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(im$95$m * t$95$2), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision] * N[(1.0 / N[(im$95$m * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 99.7

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

   8. Simplified 99.7%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 99.7%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(\mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\right), im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\right), -1\right) \cdot \frac{1}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ t_2 := im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ t_3 := im\_m \cdot t\_2\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \frac{\cos re \cdot \mathsf{fma}\left(t\_3, t\_3, -1\right)}{\mathsf{fma}\left(im\_m, t\_2, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m)))
        (t_1 (* t_0 (* 0.5 (cos re))))
        (t_2
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
           -0.16666666666666666)))
        (t_3 (* im_m t_2)))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* t_0 0.5)
      (if (<= t_1 0.1)
        (* im_m (/ (* (cos re) (fma t_3 t_3 -1.0)) (fma im_m t_2 1.0)))
        (* (- 1.0 (exp im_m)) (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = t_0 * (0.5 * cos(re));
	double t_2 = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666);
	double t_3 = im_m * t_2;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.1) {
		tmp = im_m * ((cos(re) * fma(t_3, t_3, -1.0)) / fma(im_m, t_2, 1.0));
	} else {
		tmp = (1.0 - exp(im_m)) * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(t_0 * Float64(0.5 * cos(re)))
	t_2 = Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666))
	t_3 = Float64(im_m * t_2)
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(Float64(cos(re) * fma(t_3, t_3, -1.0)) / fma(im_m, t_2, 1.0)));
	else
		tmp = Float64(Float64(1.0 - exp(im_m)) * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[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]}, Block[{t$95$2 = N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(im$95$m * t$95$2), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -1000.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[(N[Cos[re], $MachinePrecision] * N[(t$95$3 * t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(im$95$m * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 99.7

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

   8. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 99.7%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \frac{\cos re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\right), im \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\right), -1\right)}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* t_0 (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* t_0 0.5)
      (if (<= t_1 0.1)
        (*
         im_m
         (*
          (cos re)
          (fma
           (* im_m im_m)
           (fma
            im_m
            (*
             im_m
             (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
            -0.16666666666666666)
           -1.0)))
        (* (- 1.0 (exp im_m)) (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = t_0 * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= 0.1) {
		tmp = im_m * (cos(re) * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	} else {
		tmp = (1.0 - exp(im_m)) * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(t_0 * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(cos(re) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(1.0 - exp(im_m)) * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[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, -1000.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 99.7

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

   8. Simplified 99.7%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* 0.5 t_0)
      (if (<= t_1 0.1)
        (*
         im_m
         (*
          (cos re)
          (fma
           (* im_m im_m)
           (fma
            im_m
            (*
             im_m
             (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
            -0.16666666666666666)
           -1.0)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 0.1) {
		tmp = im_m * (cos(re) * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(cos(re) * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 99.7

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

   8. Simplified 99.7%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := 0.5 \cdot \cos re\\ t_2 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_1\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot -0.016666666666666666, -0.3333333333333333\right), im\_m \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* (- (exp (- im_m)) (exp im_m)) t_1)))
   (*
    im_s
    (if (<= t_2 -1000.0)
      (* 0.5 t_0)
      (if (<= t_2 0.1)
        (*
         t_1
         (fma
          (fma im_m (* im_m -0.016666666666666666) -0.3333333333333333)
          (* im_m (* im_m im_m))
          (* im_m -2.0)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = 0.5 * cos(re);
	double t_2 = (exp(-im_m) - exp(im_m)) * t_1;
	double tmp;
	if (t_2 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_2 <= 0.1) {
		tmp = t_1 * fma(fma(im_m, (im_m * -0.016666666666666666), -0.3333333333333333), (im_m * (im_m * im_m)), (im_m * -2.0));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_1)
	tmp = 0.0
	if (t_2 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_2 <= 0.1)
		tmp = Float64(t_1 * fma(fma(im_m, Float64(im_m * -0.016666666666666666), -0.3333333333333333), Float64(im_m * Float64(im_m * im_m)), Float64(im_m * -2.0)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(t$95$1 * N[(N[(im$95$m * N[(im$95$m * -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 99.5

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

   5. Simplified 99.5%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.5

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

   7. Applied egg-rr 99.5%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot -0.016666666666666666, -0.3333333333333333\right), im \cdot \left(im \cdot im\right), im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* 0.5 t_0)
      (if (<= t_1 0.1)
        (*
         im_m
         (*
          (cos re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 0.1) {
		tmp = im_m * (cos(re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(cos(re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

   5. Simplified 99.5%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := 0.5 \cdot \cos re\\ t_2 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_1\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 0.1:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right), im\_m, im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* 0.5 (cos re)))
        (t_2 (* (- (exp (- im_m)) (exp im_m)) t_1)))
   (*
    im_s
    (if (<= t_2 -1000.0)
      (* 0.5 t_0)
      (if (<= t_2 0.1)
        (* t_1 (fma (* -0.3333333333333333 (* im_m im_m)) im_m (* im_m -2.0)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = 0.5 * cos(re);
	double t_2 = (exp(-im_m) - exp(im_m)) * t_1;
	double tmp;
	if (t_2 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_2 <= 0.1) {
		tmp = t_1 * fma((-0.3333333333333333 * (im_m * im_m)), im_m, (im_m * -2.0));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(0.5 * cos(re))
	t_2 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_1)
	tmp = 0.0
	if (t_2 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_2 <= 0.1)
		tmp = Float64(t_1 * fma(Float64(-0.3333333333333333 * Float64(im_m * im_m)), im_m, Float64(im_m * -2.0)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.1], N[(t$95$1 * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

      1. distribute-rgt-in N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 99.3

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

   7. Applied egg-rr 99.3%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(-0.3333333333333333 \cdot \left(im \cdot im\right), im, im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* 0.5 t_0)
      (if (<= t_1 0.1)
        (* im_m (* (cos re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 0.1) {
		tmp = im_m * (cos(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 0.1)
		tmp = Float64(im_m * Float64(cos(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-in N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;-im\_m \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_1 -1000.0)
      (* 0.5 t_0)
      (if (<= t_1 0.1)
        (- (* im_m (cos re)))
        (* t_0 (fma -0.25 (* re re) 0.5)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = 0.5 * t_0;
	} else if (t_1 <= 0.1) {
		tmp = -(im_m * cos(re));
	} else {
		tmp = t_0 * fma(-0.25, (re * re), 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(0.5 * t_0);
	elseif (t_1 <= 0.1)
		tmp = Float64(-Float64(im_m * cos(re)));
	else
		tmp = Float64(t_0 * fma(-0.25, Float64(re * re), 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -1000.0], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.1], (-N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), N[(t$95$0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 98.7

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

   5. Simplified 98.7%

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

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

   1. Initial program 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. *-lowering-*.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 27.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;-im\_m \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -1000.0)
      (* 0.5 (- 1.0 (exp im_m)))
      (if (<= t_0 0.1)
        (- (* im_m (cos re)))
        (*
         im_m
         (*
          (fma
           (* im_m im_m)
           (fma
            im_m
            (*
             im_m
             (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
            -0.16666666666666666)
           -1.0)
          (fma
           (* re re)
           (fma
            (* re re)
            (fma (* re re) -0.001388888888888889 0.041666666666666664)
            -0.5)
           1.0))))))))

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)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else if (t_0 <= 0.1) {
		tmp = -(im_m * cos(re));
	} else {
		tmp = im_m * (fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0) * fma((re * re), fma((re * re), fma((re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.1)
		tmp = Float64(-Float64(im_m * cos(re)));
	else
		tmp = Float64(im_m * Float64(fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0) * fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[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, -1000.0], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], (-N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -1e3

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if -1e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 98.7

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

   5. Simplified 98.7%

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

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

   1. Initial program 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

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

   5. Simplified 77.5%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 77.5

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

   8. Simplified 77.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. +-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       13. unpow2 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       14. *-lowering-*.f64 59.3

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 59.3%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ t_1 := \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot 0.041666666666666664\right), 1\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot t\_1, -1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;-im\_m \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, t\_1, -1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))))
        (t_1
         (fma
          im_m
          (*
           im_m
           (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
          -0.16666666666666666)))
   (*
    im_s
    (if (<= t_0 -5000.0)
      (*
       (fma (* re re) (* re (* re 0.041666666666666664)) 1.0)
       (* im_m (fma im_m (* im_m t_1) -1.0)))
      (if (<= t_0 0.1)
        (- (* im_m (cos re)))
        (*
         im_m
         (*
          (fma (* im_m im_m) t_1 -1.0)
          (fma
           (* re re)
           (fma
            (* re re)
            (fma (* re re) -0.001388888888888889 0.041666666666666664)
            -0.5)
           1.0))))))))

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)) * (0.5 * cos(re));
	double t_1 = fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666);
	double tmp;
	if (t_0 <= -5000.0) {
		tmp = fma((re * re), (re * (re * 0.041666666666666664)), 1.0) * (im_m * fma(im_m, (im_m * t_1), -1.0));
	} else if (t_0 <= 0.1) {
		tmp = -(im_m * cos(re));
	} else {
		tmp = im_m * (fma((im_m * im_m), t_1, -1.0) * fma((re * re), fma((re * re), fma((re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	t_1 = fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666)
	tmp = 0.0
	if (t_0 <= -5000.0)
		tmp = Float64(fma(Float64(re * re), Float64(re * Float64(re * 0.041666666666666664)), 1.0) * Float64(im_m * fma(im_m, Float64(im_m * t_1), -1.0)));
	elseif (t_0 <= 0.1)
		tmp = Float64(-Float64(im_m * cos(re)));
	else
		tmp = Float64(im_m * Float64(fma(Float64(im_m * im_m), t_1, -1.0) * fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5000.0], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.1], (-N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -5e3

   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

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

   5. Simplified 91.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. *-lowering-*.f64 73.6

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.041666666666666664}, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   8. Simplified 73.6%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{24}\right) + \frac{-1}{2}\right) + 1\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \cdot im} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{24}\right) + \frac{-1}{2}\right) + 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \left(re \cdot \frac{1}{24}\right) + \frac{-1}{2}\right) + 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{24}\right) + \frac{-1}{2}, 1\right)} \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, re \cdot \left(re \cdot \frac{1}{24}\right) + \frac{-1}{2}, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\left(\left(\left(\left(im \cdot im\right) \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right) \cdot im\right)} \]

   10. Applied egg-rr 73.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot im\right)} \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{24} \cdot {re}^{2}}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{24}}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{24}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{24}\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\frac{1}{24} \cdot re\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\frac{1}{24} \cdot re\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \frac{1}{24}\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \cdot im\right) \]

       7. *-lowering-*.f64 73.6

          \[\leadsto \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot 0.041666666666666664\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot im\right) \]

   13. Simplified 73.6%

       \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot 0.041666666666666664\right)}, 1\right) \cdot \left(\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot im\right) \]

   if -5e3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 8.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 98.7

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

   5. Simplified 98.7%

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

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

   1. Initial program 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

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

   5. Simplified 77.5%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 77.5

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

   8. Simplified 77.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. +-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       13. unpow2 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       14. *-lowering-*.f64 59.3

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 59.3%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot 0.041666666666666664\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;-im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(t\_0 \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* im_m im_m)
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
           -0.16666666666666666)
          -1.0)))
   (*
    im_s
    (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
      (* im_m t_0)
      (*
       im_m
       (*
        t_0
        (fma
         (* re re)
         (fma
          (* re re)
          (fma (* re re) -0.001388888888888889 0.041666666666666664)
          -0.5)
         1.0)))))))

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= 0.0)
		tmp = Float64(im_m * t_0);
	else
		tmp = Float64(im_m * Float64(t_0 * fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0)));
	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[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]}, 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[(im$95$m * t$95$0), $MachinePrecision], N[(im$95$m * N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 77.8%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\cos re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 77.8

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

   8. Simplified 77.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. +-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{720} \cdot {re}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       13. unpow2 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       14. *-lowering-*.f64 59.9

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 59.9%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

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

Alternative 14: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      (* im_m im_m)
      (fma
       im_m
       (*
        im_m
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
       -0.16666666666666666)
      -1.0))
    (*
     (fma
      (* re re)
      (fma
       (* re re)
       (fma (* re re) -0.0006944444444444445 0.020833333333333332)
       -0.25)
      0.5)
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma im_m (* im_m -0.016666666666666666) -0.3333333333333333)
       -2.0))))))

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 = im_m * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0);
	} else {
		tmp = fma((re * re), fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25), 0.5) * (im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.016666666666666666), -0.3333333333333333), -2.0));
	}
	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(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25), 0.5) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.016666666666666666), -0.3333333333333333), -2.0)));
	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_] := 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[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 59.7

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

   8. Simplified 59.7%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      (* im_m im_m)
      (fma
       im_m
       (*
        im_m
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
       -0.16666666666666666)
      -1.0))
    (*
     (fma
      (* re re)
      (fma
       (* re re)
       (fma (* re re) -0.0006944444444444445 0.020833333333333332)
       -0.25)
      0.5)
     (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))

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 = im_m * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0);
	} else {
		tmp = fma((re * re), fma((re * re), fma((re * re), -0.0006944444444444445, 0.020833333333333332), -0.25), 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	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(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), fma(Float64(re * re), -0.0006944444444444445, 0.020833333333333332), -0.25), 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	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_] := 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[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision] + -0.25), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 61.6

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

   5. Simplified 61.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 51.9

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

   8. Simplified 51.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.4% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      (* im_m im_m)
      (fma
       im_m
       (*
        im_m
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
       -0.16666666666666666)
      -1.0))
    (-
     (fma
      (fma
       (* re re)
       (fma re (* re -0.001388888888888889) 0.041666666666666664)
       -0.5)
      (* im_m (* re re))
      im_m)))))

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= 0.0)
		tmp = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	else
		tmp = Float64(-fma(fma(Float64(re * re), fma(re, Float64(re * -0.001388888888888889), 0.041666666666666664), -0.5), Float64(im_m * Float64(re * re)), 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[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] * N[(im$95$m * N[(re * re), $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 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 6.9

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

   5. Simplified 6.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 26.5%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

   11. Simplified 26.5%

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

4. Final simplification 53.8%

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

Alternative 17: 71.4% accurate, 0.9× speedup

Math

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

FPCore

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 6.9

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

   5. Simplified 6.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 26.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{neg}\left(im \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re, \color{blue}{\frac{-1}{720} \cdot {re}^{3}}, \frac{-1}{2}\right), 1\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 26.5

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

   11. Simplified 26.5%

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

4. Final simplification 53.8%

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

Alternative 18: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      (* im_m im_m)
      (fma
       im_m
       (*
        im_m
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
       -0.16666666666666666)
      -1.0))
    (*
     (fma (* re re) -0.25 0.5)
     (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))))))

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 = im_m * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0);
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0));
	}
	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(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)));
	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_] := 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[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right) \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      9. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right)}, \frac{-1}{6}\right), -1\right) \]

      14. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)\right), \frac{-1}{6}\right), -1\right) \]

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right) \]

      18. *-lowering-*.f64 64.3

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

   8. Simplified 64.3%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 57.0

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

   8. Simplified 57.0%

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

   9. Taylor expanded in im around 0

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

       1. *-lowering-*.f64 N/A

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

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. metadata-eval N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. *-lowering-*.f64 46.6

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

   11. Simplified 46.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      im_m
      (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
      -1.0))
    (*
     (fma (* re re) -0.25 0.5)
     (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))))))

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 = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0);
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0));
	}
	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(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)));
	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_] := 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[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. exp-lowering-exp.f64 30.5

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

   5. Simplified 30.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right), -1\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}, -1\right) \]

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right) \]

      12. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right) \]

      13. *-lowering-*.f64 62.8

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

   8. Simplified 62.8%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 57.0

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

   8. Simplified 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{3}\right)} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) + \color{blue}{-2}\right)\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{3}, -2\right)}\right) \]

       8. *-lowering-*.f64 46.6

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.3333333333333333}, -2\right)\right) \]

   11. Simplified 46.6%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (*
     im_m
     (fma
      im_m
      (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
      -1.0))
    (* im_m (fma 0.5 (* re re) -1.0)))))

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 = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0);
	} else {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	}
	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(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0));
	else
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	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_] := 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[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      5. exp-lowering-exp.f64 30.5

         \[\leadsto 0.5 \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Simplified 30.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto im \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{-1}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right), -1\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}, -1\right) \]

      8. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, -1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), -1\right) \]

      10. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}\right), -1\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right) \]

      12. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right) \]

      13. *-lowering-*.f64 62.8

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right) \]

   8. Simplified 62.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \cos re}\right) \]

      4. cos-lowering-cos.f64 6.9

         \[\leadsto -im \cdot \color{blue}{\cos re} \]

   5. Simplified 6.9%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) - im} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im} \]

      5. distribute-rgt-out N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, -1\right) \]

      9. *-lowering-*.f64 19.8

         \[\leadsto im \cdot \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, -1\right) \]

   8. Simplified 19.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot -0.16666666666666666, -im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (fma (* im_m im_m) (* im_m -0.16666666666666666) (- im_m))
    (* im_m (fma 0.5 (* re re) -1.0)))))

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((im_m * im_m), (im_m * -0.16666666666666666), -im_m);
	} else {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	}
	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 = fma(Float64(im_m * im_m), Float64(im_m * -0.16666666666666666), Float64(-im_m));
	else
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	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_] := 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[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + (-im$95$m)), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      5. exp-lowering-exp.f64 30.5

         \[\leadsto 0.5 \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Simplified 30.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)} \]

      5. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right) \]

      6. *-lowering-*.f64 59.3

         \[\leadsto im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right) \]

   8. Simplified 59.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot im + -1 \cdot im} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)} \cdot im + -1 \cdot im \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\frac{-1}{6} \cdot im\right)} + -1 \cdot im \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{6} \cdot im, -1 \cdot im\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6} \cdot im, -1 \cdot im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \frac{-1}{6}}, -1 \cdot im\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \frac{-1}{6}}, -1 \cdot im\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot \frac{-1}{6}, \color{blue}{\mathsf{neg}\left(im\right)}\right) \]

      9. neg-lowering-neg.f64 59.3

         \[\leadsto \mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, \color{blue}{-im}\right) \]

   10. Applied egg-rr 59.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, -im\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \cos re}\right) \]

      4. cos-lowering-cos.f64 6.9

         \[\leadsto -im \cdot \color{blue}{\cos re} \]

   5. Simplified 6.9%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) - im} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im} \]

      5. distribute-rgt-out N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, -1\right) \]

      9. *-lowering-*.f64 19.8

         \[\leadsto im \cdot \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, -1\right) \]

   8. Simplified 19.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, -im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) 0.0)
    (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0))
    (* im_m (fma 0.5 (* re re) -1.0)))))

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 = im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0);
	} else {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	}
	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(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0));
	else
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	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_] := 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[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      5. exp-lowering-exp.f64 30.5

         \[\leadsto 0.5 \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Simplified 30.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)} \]

      5. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right) \]

      6. *-lowering-*.f64 59.3

         \[\leadsto im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right) \]

   8. Simplified 59.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \cos re}\right) \]

      4. cos-lowering-cos.f64 6.9

         \[\leadsto -im \cdot \color{blue}{\cos re} \]

   5. Simplified 6.9%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) - im} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im} \]

      5. distribute-rgt-out N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {re}^{2}, -1\right)} \]

      8. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{re \cdot re}, -1\right) \]

      9. *-lowering-*.f64 19.8

         \[\leadsto im \cdot \mathsf{fma}\left(0.5, \color{blue}{re \cdot re}, -1\right) \]

   8. Simplified 19.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) -1000.0)
    (* im_m (* (* im_m im_m) -0.16666666666666666))
    (- 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))) <= -1000.0) {
		tmp = im_m * ((im_m * im_m) * -0.16666666666666666);
	} 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 (((exp(-im_m) - exp(im_m)) * (0.5d0 * cos(re))) <= (-1000.0d0)) then
        tmp = im_m * ((im_m * im_m) * (-0.16666666666666666d0))
    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.exp(-im_m) - Math.exp(im_m)) * (0.5 * Math.cos(re))) <= -1000.0) {
		tmp = im_m * ((im_m * im_m) * -0.16666666666666666);
	} 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.exp(-im_m) - math.exp(im_m)) * (0.5 * math.cos(re))) <= -1000.0:
		tmp = im_m * ((im_m * im_m) * -0.16666666666666666)
	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 (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= -1000.0)
		tmp = Float64(im_m * Float64(Float64(im_m * im_m) * -0.16666666666666666));
	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 (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= -1000.0)
		tmp = im_m * ((im_m * im_m) * -0.16666666666666666);
	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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1000.0], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], (-im$95$m)]), $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))) < -1e3

   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      5. exp-lowering-exp.f64 76.6

         \[\leadsto 0.5 \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Simplified 76.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)} \]

      5. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right) \]

      6. *-lowering-*.f64 57.5

         \[\leadsto im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{3} \cdot \frac{-1}{6}} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \frac{-1}{6} \]

       3. unpow2 N/A

          \[\leadsto \left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \frac{-1}{6} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \frac{-1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)} \]

       9. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6}\right) \]

       10. *-lowering-*.f64 57.5

           \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot -0.16666666666666666\right) \]

   11. Simplified 57.5%

       \[\leadsto \color{blue}{im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right)} \]

   if -1e3 < (*.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 41.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \cos re}\right) \]

      4. cos-lowering-cos.f64 64.7

         \[\leadsto -im \cdot \color{blue}{\cos re} \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im\right)} \]

      2. neg-lowering-neg.f64 40.1

         \[\leadsto \color{blue}{-im} \]

   8. Simplified 40.1%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -1000:\\ \;\;\;\;im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 53.5% accurate, 18.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0))))

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 * fma(-0.16666666666666666, (im_m * im_m), -1.0));
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

   2. --lowering--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

   3. exp-lowering-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

   5. exp-lowering-exp.f64 42.0

      \[\leadsto 0.5 \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

5. Simplified 42.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

   2. sub-neg N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   3. metadata-eval N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)} \]

   5. unpow2 N/A

      \[\leadsto im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right) \]

   6. *-lowering-*.f64 57.4

      \[\leadsto im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right) \]

8. Simplified 57.4%

   \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)} \]
9. Add Preprocessing

Alternative 25: 29.8% accurate, 105.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(-im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (- im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -im_m;
}

Fortran

im\_m = 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 * Float64(-im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -im_m;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * (-im$95$m)), $MachinePrecision]

Derivation

1. Initial program 56.4%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \cos re\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \cos re}\right) \]

   4. cos-lowering-cos.f64 49.9

      \[\leadsto -im \cdot \color{blue}{\cos re} \]

5. Simplified 49.9%

   \[\leadsto \color{blue}{-im \cdot \cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot im} \]
7. Step-by-step derivation

   1. neg-mul-1 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im\right)} \]

   2. neg-lowering-neg.f64 31.1

      \[\leadsto \color{blue}{-im} \]

8. Simplified 31.1%

   \[\leadsto \color{blue}{-im} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

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 2024199 
(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))))