math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.6%

Time: 11.4s

Alternatives: 16

Speedup: 0.9×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.6% accurate, 0.6× speedup

Math

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

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 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (*
        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 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (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(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * 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[(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, (-Infinity)], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * 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 2 regimes

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

   1. Initial program 100.0%

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

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

   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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 0.004:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.001388888888888889, -0.5\right), 1\right) \cdot \left(-im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* (- (exp (- im_m)) (exp im_m)) t_0)))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* 0.5 (- 1.0 (exp im_m)))
      (if (<= t_1 0.004)
        (*
         t_0
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma im_m (* im_m -0.016666666666666666) -0.3333333333333333)
           -2.0)))
        (*
         (fma
          (* re re)
          (fma (* re re) (* (* re re) -0.001388888888888889) -0.5)
          1.0)
         (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = (exp(-im_m) - exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else if (t_1 <= 0.004) {
		tmp = t_0 * (im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.016666666666666666), -0.3333333333333333), -2.0));
	} else {
		tmp = fma((re * re), fma((re * re), ((re * re) * -0.001388888888888889), -0.5), 1.0) * -im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	elseif (t_1 <= 0.004)
		tmp = Float64(t_0 * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.016666666666666666), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(fma(Float64(re * re), fma(Float64(re * re), Float64(Float64(re * re) * -0.001388888888888889), -0.5), 1.0) * Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(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, (-Infinity)], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.004], N[(t$95$0 * 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], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * (-im$95$m)), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0040000000000000001

   1. Initial program 8.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-fma.f64 N/A

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

      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. lower-*.f64 N/A

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

      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. lower-fma.f64 N/A

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

      13. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.0040000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 91.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 91.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0.004:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \left(re \cdot re\right) \cdot -0.001388888888888889, -0.5\right), 1\right) \cdot \left(-im\right)\\ \end{array} \]
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 2024230 
(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))))