math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.9%

Time: 31.0s

Alternatives: 19

Speedup: 0.9×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.2% 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

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\\ t_1 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.12:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(t\_0, 0.027777777777777776, -1\right), \frac{1}{\mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)}, t\_0 \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* im_m (* im_m (* im_m im_m))))
        (t_1 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_1 -0.12)
      (* t_1 (* 0.5 (cos re)))
      (*
       im_m
       (*
        (cos re)
        (fma
         (fma t_0 0.027777777777777776 -1.0)
         (/ 1.0 (fma (* im_m im_m) -0.16666666666666666 1.0))
         (*
          t_0
          (fma
           (* im_m im_m)
           -0.0001984126984126984
           -0.008333333333333333)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (im_m * (im_m * im_m));
	double t_1 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_1 <= -0.12) {
		tmp = t_1 * (0.5 * cos(re));
	} else {
		tmp = im_m * (cos(re) * fma(fma(t_0, 0.027777777777777776, -1.0), (1.0 / fma((im_m * im_m), -0.16666666666666666, 1.0)), (t_0 * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333))));
	}
	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(im_m * Float64(im_m * Float64(im_m * im_m)))
	t_1 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_1 <= -0.12)
		tmp = Float64(t_1 * Float64(0.5 * cos(re)));
	else
		tmp = Float64(im_m * Float64(cos(re) * fma(fma(t_0, 0.027777777777777776, -1.0), Float64(1.0 / fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)), Float64(t_0 * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)))));
	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[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -0.12], N[(t$95$1 * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(t$95$0 * 0.027777777777777776 + -1.0), $MachinePrecision] * N[(1.0 / N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $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)) < -0.12

   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. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

      9. lift--.f64 N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -0.12 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 9.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. flip-+ N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied egg-rr 99.8%

      \[\leadsto im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right), \frac{1}{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.12:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right), \frac{1}{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}, \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.005:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.041666666666666664, -0.5\right), 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot -0.3333333333333333, im\_m, im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (cos re))) (t_1 (* (- (exp (- im_m)) (exp im_m)) t_0)))
   (*
    im_s
    (if (<= t_1 -0.005)
      (*
       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) 0.041666666666666664 -0.5)) 1.0)))
      (if (<= t_1 0.1)
        (* t_0 (fma (* (* im_m im_m) -0.3333333333333333) im_m (* im_m -2.0)))
        (*
         (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
             (fma (* im_m im_m) -0.0003968253968253968 -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 t_0 = 0.5 * cos(re);
	double t_1 = (exp(-im_m) - exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -0.005) {
		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), 0.041666666666666664, -0.5)), 1.0));
	} else if (t_1 <= 0.1) {
		tmp = t_0 * fma(((im_m * im_m) * -0.3333333333333333), im_m, (im_m * -2.0));
	} 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 * fma((im_m * im_m), -0.0003968253968253968, -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)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= -0.005)
		tmp = Float64(im_m * Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666), -1.0) * fma(re, Float64(re * fma(Float64(re * re), 0.041666666666666664, -0.5)), 1.0)));
	elseif (t_1 <= 0.1)
		tmp = Float64(t_0 * fma(Float64(Float64(im_m * im_m) * -0.3333333333333333), im_m, Float64(im_m * -2.0)));
	else
		tmp = Float64(fma(Float64(re * re), fma(re, Float64(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 * fma(Float64(im_m * im_m), -0.0003968253968253968, -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_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -0.005], N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.1], N[(t$95$0 * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * im$95$m + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445 + 0.020833333333333332), $MachinePrecision]), $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 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $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))) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 86.1%

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

   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. lower-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. lower-*.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. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 86.1

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

   8. Simplified 86.1%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 89.7

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

   11. Simplified 89.7%

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

   if -0.0050000000000000001 < (*.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 7.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.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. lower-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. lower-*.f64 99.7

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

   5. Simplified 99.7%

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

      1. lift-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.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. lower-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   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 im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 86.8

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

   5. Simplified 86.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \frac{1}{2}\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right), \frac{-1}{4}\right)}, \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{1440}} + \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{1440}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{1440}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-*.f64 97.9

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

   8. Simplified 97.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -0.005:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, 0.041666666666666664, -0.5\right), 1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.1:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot -0.3333333333333333, im, im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -0.0006944444444444445, 0.020833333333333332\right), -0.25\right), 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. 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 2024218 
(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))))