math.cos on complex, imaginary part

Percentage Accurate: 64.6% → 99.4%

Time: 15.5s

Alternatives: 20

Speedup: 2.0×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

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

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))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 (* 0.5 (sin re)))
      (* (* im_m (sin re)) (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) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (im_m * sin(re)) * fma(-0.16666666666666666, (im_m * im_m), -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(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(im_m * sin(re)) * fma(-0.16666666666666666, Float64(im_m * im_m), -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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 51.6%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 94.5%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-*.f64 N/A

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

   8. Simplified 85.3%

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

4. Final simplification 88.7%

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

Alternative 2: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (*
        re
        (fma
         (* im_m im_m)
         (fma
          im_m
          (*
           im_m
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333))
          -0.16666666666666666)
         -1.0)))
      (if (<= t_0 0.0)
        (* (* im_m (sin re)) (fma -0.16666666666666666 (* im_m im_m) -1.0))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (* (* (* im_m im_m) (* im_m im_m)) -0.0003968253968253968)
           -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)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * (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 if (t_0 <= 0.0) {
		tmp = (im_m * sin(re)) * fma(-0.16666666666666666, (im_m * im_m), -1.0);
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), (((im_m * im_m) * (im_m * im_m)) * -0.0003968253968253968), -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(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im_m * sin(re)) * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * -0.0003968253968253968), -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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 81.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 81.7%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 56.4%

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

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

   1. Initial program 29.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower-*.f64 N/A

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

   8. Simplified 99.8%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.4

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

   8. Simplified 85.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       6. lower-*.f64 76.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (*
        re
        (fma
         (* im_m im_m)
         (fma
          im_m
          (*
           im_m
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333))
          -0.16666666666666666)
         -1.0)))
      (if (<= t_0 0.0)
        (* (sin re) (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (* (* (* im_m im_m) (* im_m im_m)) -0.0003968253968253968)
           -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)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * (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 if (t_0 <= 0.0) {
		tmp = sin(re) * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), (((im_m * im_m) * (im_m * im_m)) * -0.0003968253968253968), -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(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(sin(re) * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * -0.0003968253968253968), -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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 81.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 81.7%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 56.4%

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

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

   1. Initial program 29.4%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out-- N/A

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

      12. unsub-neg N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 99.8%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.4

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

   8. Simplified 85.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       6. lower-*.f64 76.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\_m \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (*
        re
        (fma
         (* im_m im_m)
         (fma
          im_m
          (*
           im_m
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333))
          -0.16666666666666666)
         -1.0)))
      (if (<= t_0 0.0)
        (- (* im_m (sin re)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (* (* (* im_m im_m) (* im_m im_m)) -0.0003968253968253968)
           -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)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * (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 if (t_0 <= 0.0) {
		tmp = -(im_m * sin(re));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), (((im_m * im_m) * (im_m * im_m)) * -0.0003968253968253968), -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(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-Float64(im_m * sin(re)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * -0.0003968253968253968), -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[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 81.7%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 81.7%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 56.4%

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

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

   1. Initial program 29.4%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.8

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

   5. Simplified 99.8%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.4

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

   8. Simplified 85.4%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       6. lower-*.f64 76.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]

   11. Simplified 76.0%

       \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;-im \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.6× speedup

Math

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

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))))
   (*
    im_s
    (if (<= (* t_0 (* 0.5 (sin re))) (- INFINITY))
      (* t_0 (* 0.5 re))
      (*
       im_m
       (*
        (sin re)
        (fma
         im_m
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           -0.16666666666666666))
         -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);
	double tmp;
	if ((t_0 * (0.5 * sin(re))) <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * re);
	} else {
		tmp = im_m * (sin(re) * fma(im_m, (im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666)), -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(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (Float64(t_0 * Float64(0.5 * sin(re))) <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * re));
	else
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666)), -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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * 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]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-*.f64 69.1

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

   5. Simplified 69.1%

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

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

   1. Initial program 49.3%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 95.8%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 95.8%

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

4. Final simplification 88.7%

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

Alternative 6: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\left(im\_m \cdot re\right) \cdot \left(-0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(im\_m \cdot re\right), im\_m\right) \cdot \left(-re\right)\\ \end{array} \end{array} \]

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 (sin re))) -4e-8)
    (* (* im_m re) (* -0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
    (* (fma re (* -0.16666666666666666 (* im_m re)) im_m) (- re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -4e-8) {
		tmp = (im_m * re) * (-0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = fma(re, (-0.16666666666666666 * (im_m * re)), im_m) * -re;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -4e-8)
		tmp = Float64(Float64(im_m * re) * Float64(-0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(fma(re, Float64(-0.16666666666666666 * Float64(im_m * re)), im_m) * Float64(-re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-8], N[(N[(im$95$m * re), $MachinePrecision] * N[(-0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(-0.16666666666666666 * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.0000000000000001e-8

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 73.7%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 48.7

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

   8. Simplified 48.7%

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

   9. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{120} \cdot {im}^{4}\right)} \cdot \left(im \cdot re\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 48.7

          \[\leadsto \left(-0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot \left(im \cdot re\right) \]

   11. Simplified 48.7%

       \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \cdot \left(im \cdot re\right) \]

   if -4.0000000000000001e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 49.2%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 41.8%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot re\right) \cdot \left(-0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(im \cdot re\right), im\right) \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re \cdot re, im\_m \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right), -im\_m\right)\\ \end{array} \end{array} \]

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 (<= (sin re) 2e-7)
    (*
     (* re (fma (* re re) -0.08333333333333333 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)))
    (*
     re
     (fma
      (* re re)
      (* im_m (fma (* re re) -0.008333333333333333 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 (sin(re) <= 2e-7) {
		tmp = (re * fma((re * re), -0.08333333333333333, 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));
	} else {
		tmp = re * fma((re * re), (im_m * fma((re * re), -0.008333333333333333, 0.16666666666666666)), -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 (sin(re) <= 2e-7)
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(re * fma(Float64(re * re), Float64(im_m * fma(Float64(re * re), -0.008333333333333333, 0.16666666666666666)), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 2e-7], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.9999999999999999e-7

   1. Initial program 66.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. lower-*.f64 69.7

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Simplified 69.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   if 1.9999999999999999e-7 < (sin.f64 re)

   1. Initial program 50.9%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 54.4

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

   5. Simplified 54.4%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 19.4

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

   8. Simplified 19.4%

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

Alternative 8: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.0003968253968253968, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re \cdot re, im\_m \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right), -im\_m\right)\\ \end{array} \end{array} \]

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 (<= (sin re) 2e-7)
    (*
     (* re (fma (* re re) -0.08333333333333333 0.5))
     (*
      im_m
      (fma
       (* im_m im_m)
       (* (* (* im_m im_m) (* im_m im_m)) -0.0003968253968253968)
       -2.0)))
    (*
     re
     (fma
      (* re re)
      (* im_m (fma (* re re) -0.008333333333333333 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 (sin(re) <= 2e-7) {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), (((im_m * im_m) * (im_m * im_m)) * -0.0003968253968253968), -2.0));
	} else {
		tmp = re * fma((re * re), (im_m * fma((re * re), -0.008333333333333333, 0.16666666666666666)), -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 (sin(re) <= 2e-7)
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * -0.0003968253968253968), -2.0)));
	else
		tmp = Float64(re * fma(Float64(re * re), Float64(im_m * fma(Float64(re * re), -0.008333333333333333, 0.16666666666666666)), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 2e-7], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.9999999999999999e-7

   1. Initial program 66.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.7

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

   8. Simplified 92.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{2520}, -2\right)\right) \]

       6. lower-*.f64 69.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]

   11. Simplified 69.6%

       \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968, -2\right)\right) \]

   if 1.9999999999999999e-7 < (sin.f64 re)

   1. Initial program 50.9%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 54.4

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

   5. Simplified 54.4%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 19.4

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

   8. Simplified 19.4%

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

Alternative 9: 94.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 5.2 \cdot 10^{+40}:\\ \;\;\;\;im\_m \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (*
           (sin re)
           (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 (<= im_m 5200.0)
      t_0
      (if (<= im_m 5.2e+40)
        (*
         im_m
         (*
          (* re (* re (* re (+ -0.08333333333333333 (/ 0.5 (* re re))))))
          (fma (* im_m im_m) -0.3333333333333333 -2.0)))
        t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (sin(re) * 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 (im_m <= 5200.0) {
		tmp = t_0;
	} else if (im_m <= 5.2e+40) {
		tmp = im_m * ((re * (re * (re * (-0.08333333333333333 + (0.5 / (re * re)))))) * fma((im_m * im_m), -0.3333333333333333, -2.0));
	} else {
		tmp = t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666)), -1.0)))
	tmp = 0.0
	if (im_m <= 5200.0)
		tmp = t_0;
	elseif (im_m <= 5.2e+40)
		tmp = Float64(im_m * Float64(Float64(re * Float64(re * Float64(re * Float64(-0.08333333333333333 + Float64(0.5 / Float64(re * re)))))) * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	else
		tmp = t_0;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * 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]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 5200.0], t$95$0, If[LessEqual[im$95$m, 5.2e+40], N[(im$95$m * N[(N[(re * N[(re * N[(re * N[(-0.08333333333333333 + N[(0.5 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5200 or 5.2000000000000001e40 < im

   1. Initial program 61.6%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 94.9%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 94.9%

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

   if 5200 < im < 5.2000000000000001e40

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{12} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{12}\right)} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      7. lower-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{{im}^{2} \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{-1}{3}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

   8. Simplified 26.2%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)\right)\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)\right)\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} - \frac{1}{12}\right)\right)}\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)}\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(\frac{1}{2} \cdot \frac{1}{{re}^{2}} + \color{blue}{\frac{-1}{12}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       7. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{2} \cdot \frac{1}{{re}^{2}}\right)}\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       8. lower-+.f64 N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{2} \cdot \frac{1}{{re}^{2}}\right)}\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(\frac{-1}{12} + \color{blue}{\frac{\frac{1}{2} \cdot 1}{{re}^{2}}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(\frac{-1}{12} + \frac{\color{blue}{\frac{1}{2}}}{{re}^{2}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       11. lower-/.f64 N/A

           \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(\frac{-1}{12} + \color{blue}{\frac{\frac{1}{2}}{{re}^{2}}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       12. unpow2 N/A

           \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(\frac{-1}{12} + \frac{\frac{1}{2}}{\color{blue}{re \cdot re}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right)\right) \]

       13. lower-*.f64 62.9

           \[\leadsto im \cdot \left(\left(re \cdot \left(re \cdot \left(re \cdot \left(-0.08333333333333333 + \frac{0.5}{\color{blue}{re \cdot re}}\right)\right)\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right) \]

   11. Simplified 62.9%

       \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(re \cdot \left(re \cdot \left(-0.08333333333333333 + \frac{0.5}{re \cdot re}\right)\right)\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (*
     im_m
     (*
      (fma -0.16666666666666666 (* re (* re re)) re)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0)))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma
        im_m
        (*
         im_m
         (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333))
        -0.16666666666666666)
       -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 (sin(re) <= -0.01) {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma(im_m, (im_m * fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -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 (sin(re) <= -0.01)
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -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[Sin[re], $MachinePrecision], -0.01], N[(im$95$m * N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + 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[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 93.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \color{blue}{{re}^{3}} + 1 \cdot re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {re}^{3} + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 26.8

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

   8. Simplified 26.8%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 90.4%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 90.4%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 66.3%

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

Alternative 11: 57.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (*
     im_m
     (*
      im_m
      (*
       im_m
       (*
        -0.3333333333333333
        (* re (fma (* re re) -0.08333333333333333 0.5))))))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma
        im_m
        (*
         im_m
         (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333))
        -0.16666666666666666)
       -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 (sin(re) <= -0.01) {
		tmp = im_m * (im_m * (im_m * (-0.3333333333333333 * (re * fma((re * re), -0.08333333333333333, 0.5)))));
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma(im_m, (im_m * fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -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 (sin(re) <= -0.01)
		tmp = Float64(im_m * Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 * Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5))))));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333)), -0.16666666666666666), -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[Sin[re], $MachinePrecision], -0.01], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 * N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{12} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{12}\right)} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      7. lower-*.f64 27.7

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

   5. Simplified 27.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{{im}^{2} \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{-1}{3}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

   8. Simplified 25.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left({im}^{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}} \]

       2. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \cdot \frac{-1}{3} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{im \cdot \left(\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}\right)} \]

       6. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}\right)} \]

       9. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right)\right) \]

       16. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right)\right) \]

   11. Simplified 25.6%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot -0.3333333333333333\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 90.4%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   8. Simplified 90.4%

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

   9. Taylor expanded in re around 0

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

       1. lower-*.f64 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

   11. Simplified 66.3%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (*
     im_m
     (*
      im_m
      (*
       im_m
       (*
        -0.3333333333333333
        (* re (fma (* re re) -0.08333333333333333 0.5))))))
    (*
     re
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
       -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 (sin(re) <= -0.01) {
		tmp = im_m * (im_m * (im_m * (-0.3333333333333333 * (re * fma((re * re), -0.08333333333333333, 0.5)))));
	} else {
		tmp = re * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -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 (sin(re) <= -0.01)
		tmp = Float64(im_m * Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 * Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5))))));
	else
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -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[Sin[re], $MachinePrecision], -0.01], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 * N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * 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]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{12} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{12}\right)} + \frac{1}{2}\right)\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      7. lower-*.f64 27.7

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

   5. Simplified 27.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}}\right) \]

      2. associate-*r* N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + \color{blue}{{im}^{2} \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \frac{-1}{3}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) + -2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right) \]

   8. Simplified 25.4%

      \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \left({im}^{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({im}^{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}} \]

       2. cube-mult N/A

          \[\leadsto \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3} \]

       3. unpow2 N/A

          \[\leadsto \left(\left(im \cdot \color{blue}{{im}^{2}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3} \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \cdot \frac{-1}{3} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{im \cdot \left(\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}\right)} \]

       6. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{3} \cdot \left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right) \cdot \frac{-1}{3}\right)} \]

       9. associate-*r* N/A

          \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)\right)} \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)\right)\right)}\right) \]

       15. *-commutative N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right)\right) \]

       16. lower-*.f64 N/A

           \[\leadsto im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \frac{-1}{3}\right)}\right)\right) \]

   11. Simplified 25.6%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot -0.3333333333333333\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 64.3

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

   8. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.8

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 65.8

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

   10. Applied egg-rr 65.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (* re (* im_m (* re (* re 0.16666666666666666))))
    (*
     re
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
       -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 (sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = re * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -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 (sin(re) <= -0.01)
		tmp = Float64(re * Float64(im_m * Float64(re * Float64(re * 0.16666666666666666))));
	else
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -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[Sin[re], $MachinePrecision], -0.01], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * 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]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 63.7

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

   5. Simplified 63.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]

   11. Simplified 22.5%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 64.3

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

   8. Simplified 64.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.8

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 65.8

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

   10. Applied egg-rr 65.8%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (* re (* im_m (* re (* re 0.16666666666666666))))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -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 (sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -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 (sin(re) <= -0.01)
		tmp = Float64(re * Float64(im_m * Float64(re * Float64(re * 0.16666666666666666))));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -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[Sin[re], $MachinePrecision], -0.01], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 63.7

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

   5. Simplified 63.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]

   11. Simplified 22.5%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 64.3

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

   8. Simplified 64.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333, -1\right) \cdot \left(im\_m \cdot re\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (* re (* im_m (* re (* re 0.16666666666666666))))
    (*
     (fma (* im_m im_m) (* (* im_m im_m) -0.008333333333333333) -1.0)
     (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = fma((im_m * im_m), ((im_m * im_m) * -0.008333333333333333), -1.0) * (im_m * re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(re * Float64(im_m * Float64(re * Float64(re * 0.16666666666666666))));
	else
		tmp = Float64(fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * -0.008333333333333333), -1.0) * Float64(im_m * re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + -1.0), $MachinePrecision] * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 63.7

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

   5. Simplified 63.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]

   11. Simplified 22.5%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 64.3

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

   8. Simplified 64.3%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 64.1

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

   11. Simplified 64.1%

       \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right) \cdot -0.008333333333333333}, -1\right) \cdot \left(im \cdot re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right) \cdot \left(im\_m \cdot re\right)\\ \end{array} \end{array} \]

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 (<= (sin re) -0.01)
    (* re (* im_m (* re (* re 0.16666666666666666))))
    (* (fma im_m (* im_m -0.16666666666666666) -1.0) (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = fma(im_m, (im_m * -0.16666666666666666), -1.0) * (im_m * re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(re * Float64(im_m * Float64(re * Float64(re * 0.16666666666666666))));
	else
		tmp = Float64(fma(im_m, Float64(im_m * -0.16666666666666666), -1.0) * Float64(im_m * re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 63.7

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

   5. Simplified 63.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]

   11. Simplified 22.5%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Simplified 87.5%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. sub-neg N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 64.3

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

   8. Simplified 64.3%

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

   9. Taylor expanded in im around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 53.5

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

   11. Simplified 53.5%

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

Alternative 17: 34.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) 0.0001)
    (* re (* im_m (fma (* re re) 0.16666666666666666 -1.0)))
    (- (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.0001) {
		tmp = re * (im_m * fma((re * re), 0.16666666666666666, -1.0));
	} else {
		tmp = -(im_m * re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.0001)
		tmp = Float64(re * Float64(im_m * fma(Float64(re * re), 0.16666666666666666, -1.0)));
	else
		tmp = Float64(-Float64(im_m * re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 0.0001], N[(re * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(im$95$m * re), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000005e-4

   1. Initial program 66.8%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 54.9

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

   5. Simplified 54.9%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. neg-mul-1 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{6}, -1\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{6}, -1\right)\right) \]

      11. lower-*.f64 41.3

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.16666666666666666, -1\right)\right) \]

   8. Simplified 41.3%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)} \]

   if 1.00000000000000005e-4 < (sin.f64 re)

   1. Initial program 50.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 54.4

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 54.4%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot re\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot im}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      5. lower-neg.f64 8.3

         \[\leadsto re \cdot \color{blue}{\left(-im\right)} \]

   8. Simplified 8.3%

      \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.01)
    (* re (* im_m (* re (* re 0.16666666666666666))))
    (- (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = -(im_m * re);
	}
	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 (sin(re) <= (-0.01d0)) then
        tmp = re * (im_m * (re * (re * 0.16666666666666666d0)))
    else
        tmp = -(im_m * re)
    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.sin(re) <= -0.01) {
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	} else {
		tmp = -(im_m * re);
	}
	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.sin(re) <= -0.01:
		tmp = re * (im_m * (re * (re * 0.16666666666666666)))
	else:
		tmp = -(im_m * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(re * Float64(im_m * Float64(re * Float64(re * 0.16666666666666666))));
	else
		tmp = Float64(-Float64(im_m * re));
	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 (sin(re) <= -0.01)
		tmp = re * (im_m * (re * (re * 0.16666666666666666)));
	else
		tmp = -(im_m * re);
	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[Sin[re], $MachinePrecision], -0.01], N[(re * N[(im$95$m * N[(re * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(im$95$m * re), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 63.7

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot {re}^{2}\right) \cdot \frac{1}{6}} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto re \cdot \left(\color{blue}{im \cdot \left({re}^{2} \cdot \frac{1}{6}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto re \cdot \left(im \cdot \left({re}^{2} \cdot \frac{1}{6}\right) + \color{blue}{-1 \cdot im}\right) \]

      6. *-commutative N/A

         \[\leadsto re \cdot \left(im \cdot \left({re}^{2} \cdot \frac{1}{6}\right) + \color{blue}{im \cdot -1}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{6}, -1\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{6}, -1\right)\right) \]

      11. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.16666666666666666, -1\right)\right) \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto re \cdot \color{blue}{\left(\left(im \cdot {re}^{2}\right) \cdot \frac{1}{6}\right)} \]

       2. associate-*l* N/A

          \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2}\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto re \cdot \left(im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{6}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)}\right) \]

       9. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot 0.16666666666666666\right)}\right)\right) \]

   11. Simplified 22.5%

       \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 51.9

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot re\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot im}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      5. lower-neg.f64 36.1

         \[\leadsto re \cdot \color{blue}{\left(-im\right)} \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;re \cdot \left(im \cdot \left(re \cdot \left(re \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.01)
    (* 0.16666666666666666 (* im_m (* re (* re re))))
    (- (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = 0.16666666666666666 * (im_m * (re * (re * re)));
	} else {
		tmp = -(im_m * re);
	}
	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 (sin(re) <= (-0.01d0)) then
        tmp = 0.16666666666666666d0 * (im_m * (re * (re * re)))
    else
        tmp = -(im_m * re)
    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.sin(re) <= -0.01) {
		tmp = 0.16666666666666666 * (im_m * (re * (re * re)));
	} else {
		tmp = -(im_m * re);
	}
	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.sin(re) <= -0.01:
		tmp = 0.16666666666666666 * (im_m * (re * (re * re)))
	else:
		tmp = -(im_m * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(0.16666666666666666 * Float64(im_m * Float64(re * Float64(re * re))));
	else
		tmp = Float64(-Float64(im_m * re));
	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 (sin(re) <= -0.01)
		tmp = 0.16666666666666666 * (im_m * (re * (re * re)));
	else
		tmp = -(im_m * re);
	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[Sin[re], $MachinePrecision], -0.01], N[(0.16666666666666666 * N[(im$95$m * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(im$95$m * re), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 42.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 63.7

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(im \cdot {re}^{2}\right) \cdot \frac{1}{6}} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto re \cdot \left(\color{blue}{im \cdot \left({re}^{2} \cdot \frac{1}{6}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto re \cdot \left(im \cdot \left({re}^{2} \cdot \frac{1}{6}\right) + \color{blue}{-1 \cdot im}\right) \]

      6. *-commutative N/A

         \[\leadsto re \cdot \left(im \cdot \left({re}^{2} \cdot \frac{1}{6}\right) + \color{blue}{im \cdot -1}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({re}^{2} \cdot \frac{1}{6} + -1\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{6}, -1\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{6}, -1\right)\right) \]

      11. lower-*.f64 22.5

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.16666666666666666, -1\right)\right) \]

   8. Simplified 22.5%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(im \cdot {re}^{3}\right)} \]

       3. cube-mult N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \left(re \cdot \color{blue}{{re}^{2}}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \color{blue}{\left(re \cdot {re}^{2}\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

       7. lower-*.f64 22.4

          \[\leadsto 0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Simplified 22.4%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)} \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 69.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 51.9

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 51.9%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot re\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot im}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

      5. lower-neg.f64 36.1

         \[\leadsto re \cdot \color{blue}{\left(-im\right)} \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 33.2% accurate, 39.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(-im\_m \cdot re\right) \end{array} \]

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 re))))

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 * re);
}

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 * re)
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 * re);
}

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 * re)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(-Float64(im_m * re)))
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 * re);
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 * re), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 62.8%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

   4. lower-sin.f64 54.8

      \[\leadsto -im \cdot \color{blue}{\sin re} \]

5. Simplified 54.8%

   \[\leadsto \color{blue}{-im \cdot \sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot re\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot im}\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

   5. lower-neg.f64 28.8

      \[\leadsto re \cdot \color{blue}{\left(-im\right)} \]

8. Simplified 28.8%

   \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]

9. Final simplification 28.8%

   \[\leadsto -im \cdot re \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-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 = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024207 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))