math.cos on complex, imaginary part

Percentage Accurate: 64.6% → 99.4%

Time: 16.8s

Alternatives: 25

Speedup: 2.1×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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 (- INFINITY))
      (* t_0 (* 0.5 (sin 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 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * sin(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 (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * 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[t$95$0, (-Infinity)], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 53.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 96.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 96.3%

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

4. Final simplification 97.3%

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

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

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))
      (*
       (*
        re
        (fma
         (* re re)
         (fma (* re re) 0.004166666666666667 -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)))
      (if (<= t_0 5e-5)
        (*
         0.5
         (* (sin re) (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))))
        (*
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))
         (* re (fma -0.08333333333333333 (* re re) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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 if (t_0 <= 5e-5) {
		tmp = 0.5 * (sin(re) * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0)));
	} else {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	elseif (t_0 <= 5e-5)
		tmp = Float64(0.5 * Float64(sin(re) * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0))));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(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[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 69.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 72.0

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 72.0%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 35.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* (- (exp (- im_m)) (exp im_m)) t_0)))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (*
       (*
        re
        (fma
         (* re re)
         (fma (* re re) 0.004166666666666667 -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)))
      (if (<= t_1 5e-5)
        (* t_0 (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0)))
        (*
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))
         (* re (fma -0.08333333333333333 (* re re) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double t_1 = (exp(-im_m) - exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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 if (t_1 <= 5e-5) {
		tmp = t_0 * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0));
	} else {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	elseif (t_1 <= 5e-5)
		tmp = Float64(t_0 * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(t$95$0 * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 69.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 72.0

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 72.0%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 35.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

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))
      (*
       (*
        re
        (fma
         (* re re)
         (fma (* re re) 0.004166666666666667 -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)))
      (if (<= t_0 5e-5)
        (* im_m (* (sin re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))
         (* re (fma -0.08333333333333333 (* re re) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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 if (t_0 <= 5e-5) {
		tmp = im_m * (sin(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	elseif (t_0 <= 5e-5)
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(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[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 69.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 72.0

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 72.0%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 35.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-155}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -1e-155)
      (*
       im_m
       (*
        (fma
         (* re re)
         (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
         re)
        (fma
         (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
         (* im_m (* im_m (* im_m im_m)))
         (fma im_m (* im_m -0.16666666666666666) -1.0))))
      (if (<= t_0 5e-5)
        (* (- im_m) (sin re))
        (*
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))
         (* re (fma -0.08333333333333333 (* re re) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -1e-155) {
		tmp = im_m * (fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), re) * fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (im_m * (im_m * (im_m * im_m))), fma(im_m, (im_m * -0.16666666666666666), -1.0)));
	} else if (t_0 <= 5e-5) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= -1e-155)
		tmp = Float64(im_m * Float64(fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), re) * fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(im_m, Float64(im_m * -0.16666666666666666), -1.0))));
	elseif (t_0 <= 5e-5)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(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, -1e-155], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $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))) < -1.00000000000000001e-155

   1. Initial program 98.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 91.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.3%

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

   if -1.00000000000000001e-155 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 5.00000000000000024e-5

   1. Initial program 34.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 99.8%

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

   if 5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-155}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.6× speedup

Math

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

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(im_m, Float64(im_m * fma(Float64(im_m * 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[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -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 75.0

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

   5. Applied rewrites 75.0%

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

   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 54.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 94.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 94.8%

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

4. Final simplification 89.9%

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

Alternative 7: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) (- INFINITY))
    (*
     (*
      re
      (fma
       (* re re)
       (fma (* re re) 0.004166666666666667 -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)))
    (*
     im_m
     (*
      (sin 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 (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -((double) INFINITY)) {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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 = im_m * (sin(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 (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= Float64(-Inf))
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(im_m * Float64(sin(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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[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]]), $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 im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 69.0

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 72.0

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 72.0%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   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 54.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 93.8%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-7}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, -1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \]

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) 1e-7)
    (*
     im_m
     (*
      (fma re (* -0.16666666666666666 (* re re)) re)
      (/
       1.0
       (/
        1.0
        (fma
         (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
         (* im_m (* im_m (* im_m im_m)))
         (fma (* im_m im_m) -0.16666666666666666 -1.0))))))
    (*
     (*
      re
      (fma
       (* re re)
       (fma (* re re) 0.004166666666666667 -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))))))

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) <= 1e-7) {
		tmp = im_m * (fma(re, (-0.16666666666666666 * (re * re)), re) * (1.0 / (1.0 / fma(fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), (im_m * (im_m * (im_m * im_m))), fma((im_m * im_m), -0.16666666666666666, -1.0)))));
	} else {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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));
	}
	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) <= 1e-7)
		tmp = Float64(im_m * Float64(fma(re, Float64(-0.16666666666666666 * Float64(re * re)), re) * Float64(1.0 / Float64(1.0 / fma(fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(Float64(im_m * im_m), -0.16666666666666666, -1.0))))));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 1e-7], N[(im$95$m * N[(N[(re * N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 9.9999999999999995e-8

   1. Initial program 68.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 94.3%

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 68.3%

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

   if 9.9999999999999995e-8 < (sin.f64 re)

   1. Initial program 55.8%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 29.1

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 29.1

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 29.1%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-7}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, -0.16666666666666666 \cdot \left(re \cdot re\right), re\right) \cdot \frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \end{array} \end{array} \]

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) -0.005)
    (*
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
       -2.0))
     (* re (fma -0.08333333333333333 (* re re) 0.5)))
    (*
     im_m
     (*
      (fma
       (* re re)
       (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
       re)
      (fma
       (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
       (* im_m (* im_m (* im_m im_m)))
       (fma im_m (* im_m -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) <= -0.005) {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = im_m * (fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), re) * fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (im_m * (im_m * (im_m * im_m))), fma(im_m, (im_m * -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) <= -0.005)
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(im_m * Float64(fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), re) * fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(im_m, Float64(im_m * -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], -0.005], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.7

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

   8. Applied rewrites 23.7%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 93.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \]

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) -0.005)
    (*
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
       -2.0))
     (* re (fma -0.08333333333333333 (* re re) 0.5)))
    (*
     (*
      re
      (fma
       (* re re)
       (fma (* re re) 0.004166666666666667 -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))))))

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) <= -0.005) {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = (re * fma((re * re), fma((re * re), 0.004166666666666667, -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));
	}
	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) <= -0.005)
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666)), -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.7

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

   8. Applied rewrites 23.7%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 73.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

       2. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

       3. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right), \frac{-1}{3}\right)}, -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       13. sub-neg N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}\right), \frac{-1}{3}\right), -2\right)\right) \]

       16. lower-fma.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       18. lower-*.f64 74.7

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 74.7%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ t_1 := im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \frac{\mathsf{fma}\left(im\_m \cdot im\_m, t\_1 \cdot t\_1, -1\right)}{\mathsf{fma}\left(im\_m, t\_1, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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))))
        (t_1
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           -0.16666666666666666))))
   (*
    im_s
    (if (<= im_m 1.05e+27)
      t_0
      (if (<= im_m 1.6e+51)
        (*
         im_m
         (* re (/ (fma (* im_m im_m) (* t_1 t_1) -1.0) (fma im_m t_1 1.0))))
        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 t_1 = im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666);
	double tmp;
	if (im_m <= 1.05e+27) {
		tmp = t_0;
	} else if (im_m <= 1.6e+51) {
		tmp = im_m * (re * (fma((im_m * im_m), (t_1 * t_1), -1.0) / fma(im_m, t_1, 1.0)));
	} 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(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666)), -1.0)))
	t_1 = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666))
	tmp = 0.0
	if (im_m <= 1.05e+27)
		tmp = t_0;
	elseif (im_m <= 1.6e+51)
		tmp = Float64(im_m * Float64(re * Float64(fma(Float64(im_m * im_m), Float64(t_1 * t_1), -1.0) / fma(im_m, t_1, 1.0))));
	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[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, N[(im$95$s * If[LessEqual[im$95$m, 1.05e+27], t$95$0, If[LessEqual[im$95$m, 1.6e+51], N[(im$95$m * N[(re * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(im$95$m * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.04999999999999997e27 or 1.6000000000000001e51 < im

   1. Initial program 65.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 95.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 95.7%

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

   if 1.04999999999999997e27 < im < 1.6000000000000001e51

   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. Applied rewrites 4.4%

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 4.7%

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

   12. Applied rewrites 100.0%

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

Alternative 12: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \left(im\_m \cdot re\right)\\ \end{array} \end{array} \]

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) -0.005)
    (*
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
       -2.0))
     (* re (fma -0.08333333333333333 (* re re) 0.5)))
    (*
     (fma
      (* im_m im_m)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
       -0.16666666666666666)
      -1.0)
     (* im_m 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) <= -0.005) {
		tmp = (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0) * (im_m * 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) <= -0.005)
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -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], -0.005], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] + -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) < -0.0050000000000000001

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.7

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

   8. Applied rewrites 23.7%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 93.1%

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

   7. Applied rewrites 93.1%

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 74.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \left(im \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(-0.008333333333333333, im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \frac{\mathsf{fma}\left(im\_m \cdot im\_m, t\_0 \cdot t\_0, -1\right)}{\mathsf{fma}\left(im\_m, t\_0, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           -0.16666666666666666))))
   (*
    im_s
    (if (<= im_m 1.05e+27)
      (*
       im_m
       (*
        (sin re)
        (fma
         -0.008333333333333333
         (* im_m (* im_m (* im_m im_m)))
         (fma im_m (* im_m -0.16666666666666666) -1.0))))
      (if (<= im_m 9.6e+51)
        (*
         im_m
         (* re (/ (fma (* im_m im_m) (* t_0 t_0) -1.0) (fma im_m t_0 1.0))))
        (*
         (* 0.5 (sin re))
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666);
	double tmp;
	if (im_m <= 1.05e+27) {
		tmp = im_m * (sin(re) * fma(-0.008333333333333333, (im_m * (im_m * (im_m * im_m))), fma(im_m, (im_m * -0.16666666666666666), -1.0)));
	} else if (im_m <= 9.6e+51) {
		tmp = im_m * (re * (fma((im_m * im_m), (t_0 * t_0), -1.0) / fma(im_m, t_0, 1.0)));
	} else {
		tmp = (0.5 * sin(re)) * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666))
	tmp = 0.0
	if (im_m <= 1.05e+27)
		tmp = Float64(im_m * Float64(sin(re) * fma(-0.008333333333333333, Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(im_m, Float64(im_m * -0.16666666666666666), -1.0))));
	elseif (im_m <= 9.6e+51)
		tmp = Float64(im_m * Float64(re * Float64(fma(Float64(im_m * im_m), Float64(t_0 * t_0), -1.0) / fma(im_m, t_0, 1.0))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 1.05e+27], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+51], N[(im$95$m * N[(re * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(im$95$m * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 1.04999999999999997e27

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 92.8%

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

   if 1.04999999999999997e27 < im < 9.5999999999999994e51

   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. Applied rewrites 4.4%

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 4.7%

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

   12. Applied rewrites 100.0%

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

   if 9.5999999999999994e51 < im

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 14: 93.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+51}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \frac{\mathsf{fma}\left(im\_m \cdot im\_m, t\_0 \cdot t\_0, -1\right)}{\mathsf{fma}\left(im\_m, t\_0, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           -0.16666666666666666))))
   (*
    im_s
    (if (<= im_m 1.05e+27)
      (*
       im_m
       (*
        (sin re)
        (fma
         (* im_m im_m)
         (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
         -1.0)))
      (if (<= im_m 9.6e+51)
        (*
         im_m
         (* re (/ (fma (* im_m im_m) (* t_0 t_0) -1.0) (fma im_m t_0 1.0))))
        (*
         (* 0.5 (sin re))
         (*
          im_m
          (fma
           im_m
           (*
            im_m
            (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * fma((im_m * im_m), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666);
	double tmp;
	if (im_m <= 1.05e+27) {
		tmp = im_m * (sin(re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else if (im_m <= 9.6e+51) {
		tmp = im_m * (re * (fma((im_m * im_m), (t_0 * t_0), -1.0) / fma(im_m, t_0, 1.0)));
	} else {
		tmp = (0.5 * sin(re)) * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -0.16666666666666666))
	tmp = 0.0
	if (im_m <= 1.05e+27)
		tmp = Float64(im_m * Float64(sin(re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	elseif (im_m <= 9.6e+51)
		tmp = Float64(im_m * Float64(re * Float64(fma(Float64(im_m * im_m), Float64(t_0 * t_0), -1.0) / fma(im_m, t_0, 1.0))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 1.05e+27], N[(im$95$m * N[(N[Sin[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], If[LessEqual[im$95$m, 9.6e+51], N[(im$95$m * N[(re * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(im$95$m * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 1.04999999999999997e27

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. Applied rewrites 92.8%

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

   if 1.04999999999999997e27 < im < 9.5999999999999994e51

   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. Applied rewrites 4.4%

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 4.7%

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

   12. Applied rewrites 100.0%

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

   if 9.5999999999999994e51 < im

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 15: 58.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \left(im\_m \cdot re\right)\\ \end{array} \end{array} \]

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) -0.005)
    (*
     (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))
     (* re (fma -0.08333333333333333 (* re re) 0.5)))
    (*
     (fma
      (* im_m im_m)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
       -0.16666666666666666)
      -1.0)
     (* im_m 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) <= -0.005) {
		tmp = (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0) * (im_m * 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) <= -0.005)
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -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], -0.005], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] + -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) < -0.0050000000000000001

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 22.5

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

   8. Applied rewrites 22.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 93.1%

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

   7. Applied rewrites 93.1%

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 74.3%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \left(im \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.005)
    (*
     (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))
     (* re (fma -0.08333333333333333 (* re re) 0.5)))
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
       -2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0)) * (re * fma(-0.08333333333333333, (re * re), 0.5));
	} else {
		tmp = (0.5 * re) * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)) * Float64(re * fma(-0.08333333333333333, Float64(re * re), 0.5)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 22.5

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

   8. Applied rewrites 22.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.005)
    (* (* re (* im_m re)) (* re 0.16666666666666666))
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
       -2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = (re * (im_m * re)) * (re * 0.16666666666666666);
	} else {
		tmp = (0.5 * re) * (im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * re)) * Float64(re * 0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.005)
    (* (* re (* im_m re)) (* re 0.16666666666666666))
    (*
     (* 0.5 re)
     (*
      im_m
      (fma (* im_m im_m) (* (* im_m im_m) -0.016666666666666666) -2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = (re * (im_m * re)) * (re * 0.16666666666666666);
	} else {
		tmp = (0.5 * re) * (im_m * fma((im_m * im_m), ((im_m * im_m) * -0.016666666666666666), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * re)) * Float64(re * 0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * -0.016666666666666666), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 73.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot -0.016666666666666666, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

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) -0.005)
    (* (* 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) <= -0.005) {
		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) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * re)) * Float64(re * 0.16666666666666666));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(im_m, Float64(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], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $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 * -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 74.3%

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

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 72.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.005)
    (* (* re (* im_m re)) (* re 0.16666666666666666))
    (* (* 0.5 re) (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = (re * (im_m * re)) * (re * 0.16666666666666666);
	} else {
		tmp = (0.5 * re) * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * re)) * Float64(re * 0.16666666666666666));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 66.3

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

   8. Applied rewrites 66.3%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 50.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\\ \end{array} \end{array} \]

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) -0.005)
    (* (* re (* im_m re)) (* re 0.16666666666666666))
    (* im_m (* 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 tmp;
	if (sin(re) <= -0.005) {
		tmp = (re * (im_m * re)) * (re * 0.16666666666666666);
	} else {
		tmp = im_m * (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)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * re)) * Float64(re * 0.16666666666666666));
	else
		tmp = Float64(im_m * Float64(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_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   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. Applied rewrites 93.1%

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 74.3%

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

   11. Taylor expanded in im around 0

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

   13. Applied rewrites 63.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot 0.16666666666666666, -1\right) \cdot \left(im\_m \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

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) 4e-5)
    (* (fma re (* re 0.16666666666666666) -1.0) (* 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 (sin(re) <= 4e-5) {
		tmp = fma(re, (re * 0.16666666666666666), -1.0) * (im_m * 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) <= 4e-5)
		tmp = Float64(fma(re, Float64(re * 0.16666666666666666), -1.0) * Float64(im_m * re));
	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], 4e-5], N[(N[(re * N[(re * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision], (-N[(im$95$m * re), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 4.00000000000000033e-5

   1. Initial program 68.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.8%

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

   10. Applied rewrites 42.8%

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

   if 4.00000000000000033e-5 < (sin.f64 re)

   1. Initial program 55.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 23.0%

      \[\leadsto -im \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-5}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(re \cdot re, 0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

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) 4e-5)
    (* 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) <= 4e-5) {
		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) <= 4e-5)
		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], 4e-5], 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) < 4.00000000000000033e-5

   1. Initial program 68.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.8%

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

   if 4.00000000000000033e-5 < (sin.f64 re)

   1. Initial program 55.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 23.0%

      \[\leadsto -im \cdot re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 35.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im\_m \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m \cdot re\\ \end{array} \end{array} \]

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) -0.005)
    (* (* 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) <= -0.005) {
		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) <= (-0.005d0)) 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) <= -0.005) {
		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) <= -0.005:
		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) <= -0.005)
		tmp = Float64(Float64(re * Float64(im_m * 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) <= -0.005)
		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], -0.005], N[(N[(re * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] * N[(re * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], (-N[(im$95$m * re), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 19.5%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 71.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 46.1%

      \[\leadsto -im \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;\left(re \cdot \left(im \cdot re\right)\right) \cdot \left(re \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 33.5% 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 65.9%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
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 54.6

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]
7. Step-by-step derivation

8. Applied rewrites 37.1%

   \[\leadsto -im \cdot re \]
9. Add Preprocessing

Developer Target 1: 99.7% 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 2024233 
(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))))