math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.9%

Time: 16.2s

Alternatives: 26

Speedup: 0.7×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 65.2% 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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.2) {
		tmp = fma((t_0 * sin(re)), 0.5, (exp(im_m) * (sin(re) * -0.5)));
	} else {
		tmp = (sin(re) * 0.5) * fma(fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), (im_m * (im_m * im_m)), (im_m * -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.2)
		tmp = fma(Float64(t_0 * sin(re)), 0.5, Float64(exp(im_m) * Float64(sin(re) * Float64(-0.5))));
	else
		tmp = Float64(Float64(sin(re) * 0.5) * fma(fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), Float64(im_m * Float64(im_m * im_m)), Float64(im_m * -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(t$95$0 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[Exp[im$95$m], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. sin-lowering-sin.f64 N/A

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

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

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

      16. exp-lowering-exp.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 60.2%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.8

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

   5. Simplified 93.8%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 93.8

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

   7. Applied egg-rr 93.8%

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

4. Final simplification 95.4%

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

Alternative 2: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right)\\ t_1 := \sin re \cdot 0.5\\ t_2 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_1\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+93}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(t\_0, im\_m \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, t\_0, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333))
        (t_1 (* (sin re) 0.5))
        (t_2 (* (- (exp (- im_m)) (exp im_m)) t_1)))
   (*
    im_s
    (if (<= t_2 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_2 1e+93)
        (* t_1 (fma t_0 (* im_m (* im_m im_m)) (* im_m -2.0)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (* im_m (fma (* im_m im_m) t_0 -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 = fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333);
	double t_1 = sin(re) * 0.5;
	double t_2 = (exp(-im_m) - exp(im_m)) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_2 <= 1e+93) {
		tmp = t_1 * fma(t_0, (im_m * (im_m * im_m)), (im_m * -2.0));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), t_0, -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 = fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333)
	t_1 = Float64(sin(re) * 0.5)
	t_2 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_2 <= 1e+93)
		tmp = Float64(t_1 * fma(t_0, Float64(im_m * Float64(im_m * im_m)), Float64(im_m * -2.0)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), t_0, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+93], N[(t$95$1 * N[(t$95$0 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 97.5

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

   5. Simplified 97.5%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 97.6

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

   7. Applied egg-rr 97.6%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), im \cdot \left(im \cdot im\right), im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 1e+93)
        (*
         im_m
         (*
          (sin re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 1e+93) {
		tmp = im_m * (sin(re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 1e+93)
		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)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], 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], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

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

   3. Taylor expanded in im around 0

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

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

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

      2. +-commutative N/A

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

   5. Simplified 97.5%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im\_m \cdot \left(im\_m \cdot im\_m\right), -0.16666666666666666, -im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 1e+93)
        (* (sin re) (fma (* im_m (* im_m im_m)) -0.16666666666666666 (- im_m)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 1e+93) {
		tmp = sin(re) * fma((im_m * (im_m * im_m)), -0.16666666666666666, -im_m);
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 1e+93)
		tmp = Float64(sin(re) * fma(Float64(im_m * Float64(im_m * im_m)), -0.16666666666666666, Float64(-im_m)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + (-im$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 97.3%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

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

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

      9. neg-lowering-neg.f64 97.3

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

   7. Applied egg-rr 97.3%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 1e+93)
        (* im_m (* (sin re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 1e+93) {
		tmp = im_m * (sin(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), 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(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 1e+93)
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), 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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], 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[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 97.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. sin-lowering-sin.f64 97.3

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

   7. Applied egg-rr 97.3%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 1e+93)
        (* (sin re) (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (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 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 1e+93) {
		tmp = sin(re) * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), 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(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 1e+93)
		tmp = Float64(sin(re) * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), 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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 97.3%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;-im\_m \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 1e+93)
        (- (* im_m (sin re)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 1e+93) {
		tmp = -(im_m * sin(re));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 1e+93)
		tmp = Float64(-Float64(im_m * sin(re)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), 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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+93], (-N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. sin-lowering-sin.f64 96.7

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

   5. Simplified 96.7%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;-im \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(re \cdot 0.008333333333333333\right)\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+93}:\\ \;\;\;\;-im\_m \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (*
        (fma (* re re) (* re (* re (* re 0.008333333333333333))) 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 1e+93)
        (- (* im_m (sin re)))
        (*
         (* re (fma (* re re) -0.08333333333333333 0.5))
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (sin(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * (fma((re * re), (re * (re * (re * 0.008333333333333333))), 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 <= 1e+93) {
		tmp = -(im_m * sin(re));
	} else {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * Float64(fma(Float64(re * re), Float64(re * Float64(re * Float64(re * 0.008333333333333333))), 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 <= 1e+93)
		tmp = Float64(-Float64(im_m * sin(re)));
	else
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), 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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(re * 0.008333333333333333), $MachinePrecision]), $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, 1e+93], (-N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 85.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 60.3

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

   8. Simplified 60.3%

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

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{120} \cdot {re}^{3}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{3} \cdot \frac{1}{120}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 60.3

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

   11. Simplified 60.3%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.00000000000000004e93

   1. Initial program 39.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. sin-lowering-sin.f64 96.7

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

   5. Simplified 96.7%

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

   if 1.00000000000000004e93 < (*.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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 79.8

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

   5. Simplified 79.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.5

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

   8. Simplified 66.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(re \cdot 0.008333333333333333\right)\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq 10^{+93}:\\ \;\;\;\;-im \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = sin(re) * 0.5;
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * fma(fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), (im_m * (im_m * im_m)), (im_m * -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(sin(re) * 0.5)
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * fma(fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), Float64(im_m * Float64(im_m * im_m)), Float64(im_m * -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 60.2%

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

   3. Taylor expanded in im around 0

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

      1. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 93.8

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

   5. Simplified 93.8%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 93.8

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

   7. Applied egg-rr 93.8%

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

4. Final simplification 95.4%

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

Alternative 10: 89.5% accurate, 0.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5)) (- INFINITY))
    (* (* re 0.5) (- 1.0 (exp im_m)))
    (*
     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 tmp;
	if (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5)) <= -((double) INFINITY)) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} 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)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5)) <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(im_m * Float64(sin(re) * 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)));
	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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $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] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] + -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 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. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 31.2%

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

   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 61.6%

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

   3. Taylor expanded in im around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 93.2%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 93.2

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

   8. Simplified 93.2%

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

4. Final simplification 78.7%

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

Alternative 11: 48.0% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5)) 0.0)
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
        -0.16666666666666666)
       -1.0)))
    (*
     (- im_m)
     (fma
      (fma
       (* re re)
       (fma -0.0001984126984126984 (* re re) 0.008333333333333333)
       -0.16666666666666666)
      (* re (* re re))
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5)) <= 0.0) {
		tmp = im_m * (re * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0));
	} else {
		tmp = -im_m * fma(fma((re * re), fma(-0.0001984126984126984, (re * re), 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return im_s * tmp;
}

Julia

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

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(re * 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]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(N[(N[(re * re), $MachinePrecision] * N[(-0.0001984126984126984 * N[(re * re), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $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))) < 0.0

   1. Initial program 58.6%

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

   3. Taylor expanded in im around 0

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

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

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

   5. Simplified 95.1%

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

   6. Taylor expanded in im around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 95.1

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

   8. Simplified 95.1%

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

   9. Taylor expanded in re around 0

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

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

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

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

       6. unpow2 N/A

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

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

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

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

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

       11. unpow2 N/A

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

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

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 57.8

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

   11. Simplified 57.8%

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

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

   1. Initial program 98.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

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

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

   8. Simplified 28.1%

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

4. Final simplification 49.0%

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

Alternative 12: 55.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \end{array} \end{array} \]

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)) (* (sin re) 0.5)) -5e-33)
    (* -0.008333333333333333 (* re (* im_m (* (* im_m im_m) (* im_m im_m)))))
    (*
     re
     (*
      (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
      (fma re (* re -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)) * (sin(re) * 0.5)) <= -5e-33) {
		tmp = -0.008333333333333333 * (re * (im_m * ((im_m * im_m) * (im_m * im_m))));
	} else {
		tmp = re * ((im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma(re, (re * -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(sin(re) * 0.5)) <= -5e-33)
		tmp = Float64(-0.008333333333333333 * Float64(re * Float64(im_m * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))));
	else
		tmp = Float64(re * Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(re, Float64(re * -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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -5e-33], N[(-0.008333333333333333 * N[(re * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * -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))) < -5.00000000000000028e-33

   1. Initial program 99.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 82.5

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

   5. Simplified 82.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

   8. Simplified 52.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 52.3

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

   10. Applied egg-rr 52.3%

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

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{120} \cdot \left({im}^{5} \cdot re\right)} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{120} \cdot \left({im}^{5} \cdot re\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{120} \cdot \color{blue}{\left(re \cdot {im}^{5}\right)} \]

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

          \[\leadsto \frac{-1}{120} \cdot \color{blue}{\left(re \cdot {im}^{5}\right)} \]

       4. metadata-eval N/A

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

       5. pow-plus N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 56.9

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

   13. Simplified 56.9%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 61.2%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 86.3%

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

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

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

      9. neg-lowering-neg.f64 86.3

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

   7. Applied egg-rr 86.3%

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

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]

      2. associate--l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)} + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      4. distribute-lft1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

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

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      8. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. sub-neg N/A

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

      16. unpow3 N/A

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

      17. unpow2 N/A

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

      18. associate-*r* N/A

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

      19. mul-1-neg N/A

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

   10. Simplified 57.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \end{array} \end{array} \]

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)) (* (sin re) 0.5)) -5e-33)
    (* -0.008333333333333333 (* re (* im_m (* (* im_m im_m) (* im_m im_m)))))
    (* re (* im_m (fma 0.16666666666666666 (* re re) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5)) <= -5e-33) {
		tmp = -0.008333333333333333 * (re * (im_m * ((im_m * im_m) * (im_m * im_m))));
	} else {
		tmp = re * (im_m * fma(0.16666666666666666, (re * re), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5)) <= -5e-33)
		tmp = Float64(-0.008333333333333333 * Float64(re * Float64(im_m * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))));
	else
		tmp = Float64(re * Float64(im_m * fma(0.16666666666666666, Float64(re * re), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -5e-33], N[(-0.008333333333333333 * N[(re * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(0.16666666666666666 * N[(re * re), $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))) < -5.00000000000000028e-33

   1. Initial program 99.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.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. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 82.5

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

   5. Simplified 82.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

   8. Simplified 52.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 52.3

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

   10. Applied egg-rr 52.3%

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

   11. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\frac{-1}{120} \cdot \left({im}^{5} \cdot re\right)} \]
   12. Step-by-step derivation

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

          \[\leadsto \color{blue}{\frac{-1}{120} \cdot \left({im}^{5} \cdot re\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{-1}{120} \cdot \color{blue}{\left(re \cdot {im}^{5}\right)} \]

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

          \[\leadsto \frac{-1}{120} \cdot \color{blue}{\left(re \cdot {im}^{5}\right)} \]

       4. metadata-eval N/A

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

       5. pow-plus N/A

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

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

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

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

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 56.9

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

   13. Simplified 56.9%

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

   if -5.00000000000000028e-33 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 61.2%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 63.3

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

   5. Simplified 63.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.6

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

   8. Simplified 43.6%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* (sin re) 0.5)) -5e-33)
    (* re (* (* im_m (* im_m im_m)) -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 (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5)) <= -5e-33) {
		tmp = re * ((im_m * (im_m * im_m)) * -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 (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5d0)) <= (-5d-33)) then
        tmp = re * ((im_m * (im_m * im_m)) * (-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.exp(-im_m) - Math.exp(im_m)) * (Math.sin(re) * 0.5)) <= -5e-33) {
		tmp = re * ((im_m * (im_m * im_m)) * -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.exp(-im_m) - math.exp(im_m)) * (math.sin(re) * 0.5)) <= -5e-33:
		tmp = re * ((im_m * (im_m * im_m)) * -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 (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(sin(re) * 0.5)) <= -5e-33)
		tmp = Float64(re * Float64(Float64(im_m * Float64(im_m * im_m)) * -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 (((exp(-im_m) - exp(im_m)) * (sin(re) * 0.5)) <= -5e-33)
		tmp = re * ((im_m * (im_m * im_m)) * -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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -5e-33], N[(re * N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -5.00000000000000028e-33

   1. Initial program 99.8%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

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

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

      14. sin-lowering-sin.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. *-commutative N/A

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

   5. Simplified 69.7%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

         \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sin re \cdot \frac{1}{\color{blue}{\frac{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

      7. sub-neg N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(-1\right)\right)}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{\left(im \cdot im\right) \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(-1\right)\right)}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\left(im \cdot im\right) \cdot \frac{-1}{6} + \color{blue}{1}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6}, 1\right)}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}{\color{blue}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

   7. Applied egg-rr 25.5%

      \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right) \cdot im}}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), \frac{1}{36}, -1\right) \cdot im}} \]
   9. Step-by-step derivation

   10. Simplified 20.1%

       \[\leadsto \color{blue}{re} \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right) \cdot im}} \]

   11. Taylor expanded in im around inf

       \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       6. *-lowering-*.f64 44.6

          \[\leadsto re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   13. Simplified 44.6%

       \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} \]

   if -5.00000000000000028e-33 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 61.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. sin-lowering-sin.f64 63.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 63.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(im \cdot \color{blue}{re}\right) \]
   7. Step-by-step derivation

   8. Simplified 39.6%

      \[\leadsto -im \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right) \leq -5 \cdot 10^{-33}:\\ \;\;\;\;re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-36}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, t\_0, -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(t\_0, im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)))
   (*
    im_s
    (if (<= (sin re) 2e-36)
      (*
       im_m
       (*
        (fma (* im_m im_m) (fma (* im_m im_m) t_0 -0.16666666666666666) -1.0)
        (fma re (* re (* re -0.16666666666666666)) re)))
      (*
       im_m
       (*
        (fma
         (* re re)
         (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
         re)
        (fma t_0 (* im_m (* im_m (* im_m im_m))) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333);
	double tmp;
	if (sin(re) <= 2e-36) {
		tmp = im_m * (fma((im_m * im_m), fma((im_m * im_m), t_0, -0.16666666666666666), -1.0) * fma(re, (re * (re * -0.16666666666666666)), re));
	} else {
		tmp = im_m * (fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), re) * fma(t_0, (im_m * (im_m * (im_m * im_m))), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)
	tmp = 0.0
	if (sin(re) <= 2e-36)
		tmp = Float64(im_m * Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), t_0, -0.16666666666666666), -1.0) * fma(re, Float64(re * Float64(re * -0.16666666666666666)), re)));
	else
		tmp = Float64(im_m * Float64(fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), re) * fma(t_0, Float64(im_m * Float64(im_m * Float64(im_m * im_m))), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 2e-36], N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $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[(t$95$0 * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.9999999999999999e-36

   1. Initial program 76.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 92.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      16. *-lowering-*.f64 92.4

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 92.4%

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. *-rgt-identity N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-lowering-*.f64 71.0

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 71.0%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   if 1.9999999999999999e-36 < (sin.f64 re)

   1. Initial program 57.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 88.8%

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       2. distribute-rgt-in N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \left(\left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + 1 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       4. *-lft-identity N/A

          \[\leadsto im \cdot \left(\left({re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       9. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       11. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       12. unpow2 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

       13. *-lowering-*.f64 26.0

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]

   11. Simplified 26.0%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-36}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\right)\\ \end{array} \end{array} \]

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) 4e-23)
    (*
     im_m
     (*
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
        -0.16666666666666666)
       -1.0)
      (fma re (* re (* re -0.16666666666666666)) re)))
    (*
     im_m
     (*
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0)
      (fma
       (* re re)
       (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
       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-23) {
		tmp = im_m * (fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0) * fma(re, (re * (re * -0.16666666666666666)), re));
	} else {
		tmp = im_m * (fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0) * fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), 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-23)
		tmp = Float64(im_m * 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) * fma(re, Float64(re * Float64(re * -0.16666666666666666)), re)));
	else
		tmp = Float64(im_m * Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0) * fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), 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-23], N[(im$95$m * 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[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(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] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 3.99999999999999984e-23

   1. Initial program 76.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      16. *-lowering-*.f64 92.5

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 92.5%

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. *-rgt-identity N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-lowering-*.f64 71.4

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 71.4%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   if 3.99999999999999984e-23 < (sin.f64 re)

   1. Initial program 57.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto im \cdot \left(\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto im \cdot \left(\left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      13. *-lowering-*.f64 24.1

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   8. Simplified 24.1%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)}\right) \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)}\right) \]

       4. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

       6. sub-neg N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

       9. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

       10. unpow2 N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, re \cdot re, \frac{-1}{6}\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

       11. *-lowering-*.f64 23.0

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 23.0%

       \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right) \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\\ \end{array} \end{array} \]

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) 4e-23)
    (*
     im_m
     (*
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
        -0.16666666666666666)
       -1.0)
      (fma re (* re (* re -0.16666666666666666)) re)))
    (*
     (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
     (fma
      (* re re)
      (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
      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-23) {
		tmp = im_m * (fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0) * fma(re, (re * (re * -0.16666666666666666)), re));
	} else {
		tmp = (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), 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-23)
		tmp = Float64(im_m * 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) * fma(re, Float64(re * Float64(re * -0.16666666666666666)), re)));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), 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-23], N[(im$95$m * 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[(re * N[(re * N[(re * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 3.99999999999999984e-23

   1. Initial program 76.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      16. *-lowering-*.f64 92.5

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 92.5%

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       2. distribute-lft-in N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       3. *-rgt-identity N/A

          \[\leadsto im \cdot \left(\left(re \cdot \left(\frac{-1}{6} \cdot {re}^{2}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot {re}^{2}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{{re}^{2} \cdot \frac{-1}{6}}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       8. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       10. *-commutative N/A

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       11. *-lowering-*.f64 71.4

           \[\leadsto im \cdot \left(\mathsf{fma}\left(re, re \cdot \color{blue}{\left(re \cdot -0.16666666666666666\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 71.4%

       \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   if 3.99999999999999984e-23 < (sin.f64 re)

   1. Initial program 57.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + \color{blue}{re}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      13. *-lowering-*.f64 22.9

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \]

   8. Simplified 22.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right) \cdot \mathsf{fma}\left(re, re \cdot \left(re \cdot -0.16666666666666666\right), re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\\ \end{array} \end{array} \]

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) 4e-23)
    (*
     im_m
     (*
      re
      (*
       (fma re (* re -0.16666666666666666) 1.0)
       (fma
        (* (* im_m im_m) (* im_m im_m))
        (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
        -1.0))))
    (*
     (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
     (fma
      (* re re)
      (* re (fma 0.008333333333333333 (* re re) -0.16666666666666666))
      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-23) {
		tmp = im_m * (re * (fma(re, (re * -0.16666666666666666), 1.0) * fma(((im_m * im_m) * (im_m * im_m)), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -1.0)));
	} else {
		tmp = (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma((re * re), (re * fma(0.008333333333333333, (re * re), -0.16666666666666666)), 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-23)
		tmp = Float64(im_m * Float64(re * Float64(fma(re, Float64(re * -0.16666666666666666), 1.0) * fma(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -1.0))));
	else
		tmp = Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(Float64(re * re), Float64(re * fma(0.008333333333333333, Float64(re * re), -0.16666666666666666)), 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-23], N[(im$95$m * N[(re * N[(N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * N[(re * N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 3.99999999999999984e-23

   1. Initial program 76.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 92.1%

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right) + {im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right) + {im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right) - 1\right)\right)} \]

       2. associate--l+ N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right) + \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)}\right) \]

       3. associate-*r* N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)} + \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       4. distribute-lft1-in N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)}\right) \]

       5. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)}\right) \]

       7. +-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto im \cdot \left(re \cdot \left(\left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)} + 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot re, 1\right)} \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)\right) \]

   11. Simplified 71.4%

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)\right)} \]

   if 3.99999999999999984e-23 < (sin.f64 re)

   1. Initial program 57.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + \color{blue}{re}\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re\right)} + re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot re}, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \left(\frac{1}{120} \cdot {re}^{2} + \color{blue}{\frac{-1}{6}}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {re}^{2}, \frac{-1}{6}\right)} \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\frac{1}{120}, \color{blue}{re \cdot re}, \frac{-1}{6}\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]

      13. *-lowering-*.f64 22.9

          \[\leadsto \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, \color{blue}{re \cdot re}, -0.16666666666666666\right) \cdot re, re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \]

   8. Simplified 22.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot re, re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 4 \cdot 10^{-23}:\\ \;\;\;\;im \cdot \left(re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re \cdot re, re \cdot \mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.05)
    (*
     (* re (fma (* re re) -0.08333333333333333 0.5))
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
       -2.0)))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
        -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = (re * fma((re * re), -0.08333333333333333, 0.5)) * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.05)
		tmp = Float64(Float64(re * fma(Float64(re * re), -0.08333333333333333, 0.5)) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)));
	else
		tmp = Float64(im_m * Float64(re * 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)));
	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], -0.05], N[(N[(re * N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 65.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.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. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{60} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{60}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{60} + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. *-lowering-*.f64 89.2

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   5. Simplified 89.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      6. *-lowering-*.f64 25.6

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   8. Simplified 25.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 72.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 91.9%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      16. *-lowering-*.f64 91.9

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 91.9%

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]

       3. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

       13. sub-neg N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

       14. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

       17. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       18. *-lowering-*.f64 66.1

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 66.1%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 57.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.05)
    (*
     re
     (*
      (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
      (fma re (* re -0.16666666666666666) 1.0)))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
        -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = re * ((im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma(re, (re * -0.16666666666666666), 1.0));
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.05)
		tmp = Float64(re * Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(re, Float64(re * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(im_m * Float64(re * 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)));
	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], -0.05], N[(re * N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 65.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) + im \cdot -1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)} + im \cdot -1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6}} + im \cdot -1\right) \]

      4. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{-1 \cdot im}\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), \frac{-1}{6}, \mathsf{neg}\left(im\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \color{blue}{\left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      9. neg-lowering-neg.f64 85.6

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, \color{blue}{-im}\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, -im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]

      2. associate--l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)} + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      4. distribute-lft1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      8. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot re, 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      15. sub-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3} + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      16. unpow3 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \left(\color{blue}{{im}^{2}} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right)\right) \]

   10. Simplified 25.6%

       \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 72.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 91.9%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

      16. *-lowering-*.f64 91.9

          \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 91.9%

      \[\leadsto im \cdot \left(\sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)}\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \]

       3. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)}\right) \]

       6. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}, -1\right)\right) \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

       10. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

       13. sub-neg N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{-1}{6}\right), -1\right)\right) \]

       14. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), \frac{-1}{6}\right), -1\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{5040} + \color{blue}{\frac{-1}{120}}, \frac{-1}{6}\right), -1\right)\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{-1}{120}\right)}, \frac{-1}{6}\right), -1\right)\right) \]

       17. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{-1}{120}\right), \frac{-1}{6}\right), -1\right)\right) \]

       18. *-lowering-*.f64 66.1

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right) \]

   11. Simplified 66.1%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 57.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right), \mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)\\ \end{array} \end{array} \]

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) -0.05)
    (*
     re
     (*
      (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
      (fma re (* re -0.16666666666666666) 1.0)))
    (*
     im_m
     (*
      re
      (fma
       (* (* im_m im_m) (* im_m im_m))
       (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
       -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) <= -0.05) {
		tmp = re * ((im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma(re, (re * -0.16666666666666666), 1.0));
	} else {
		tmp = im_m * (re * fma(((im_m * im_m) * (im_m * im_m)), fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), -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) <= -0.05)
		tmp = Float64(re * Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(re, Float64(re * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)), fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), -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], -0.05], N[(re * N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 65.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) + im \cdot -1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)} + im \cdot -1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6}} + im \cdot -1\right) \]

      4. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{-1 \cdot im}\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), \frac{-1}{6}, \mathsf{neg}\left(im\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \color{blue}{\left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      9. neg-lowering-neg.f64 85.6

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, \color{blue}{-im}\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, -im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]

      2. associate--l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)} + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      4. distribute-lft1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      8. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot re, 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      15. sub-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3} + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      16. unpow3 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \left(\color{blue}{{im}^{2}} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right)\right) \]

   10. Simplified 25.6%

       \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 72.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      3. associate-+r+ N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      5. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. Simplified 91.9%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 91.7%

      \[\leadsto im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)\right)} \]

       3. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{-1}\right)\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{4}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left({im}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       7. pow-sqr N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot {im}^{2}}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot {im}^{2}}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}, \frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}, -1\right)\right) \]

       13. sub-neg N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}, -1\right)\right) \]

       14. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), -1\right)\right) \]

       15. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{5040} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), -1\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \color{blue}{im \cdot \left(im \cdot \frac{-1}{5040}\right)} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), -1\right)\right) \]

       17. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right), -1\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), im \cdot \left(\frac{-1}{5040} \cdot im\right) + \color{blue}{\frac{-1}{120}}, -1\right)\right) \]

       19. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \color{blue}{\mathsf{fma}\left(im, \frac{-1}{5040} \cdot im, \frac{-1}{120}\right)}, -1\right)\right) \]

       20. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{5040}}, \frac{-1}{120}\right), -1\right)\right) \]

       21. *-lowering-*.f64 66.0

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, \color{blue}{im \cdot -0.0001984126984126984}, -0.008333333333333333\right), -1\right)\right) \]

   11. Simplified 66.0%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 56.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(0.5 \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)\\ \end{array} \end{array} \]

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) -0.05)
    (*
     re
     (*
      (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))
      (fma re (* re -0.16666666666666666) 1.0)))
    (*
     re
     (*
      im_m
      (*
       0.5
       (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) <= -0.05) {
		tmp = re * ((im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0)) * fma(re, (re * -0.16666666666666666), 1.0));
	} else {
		tmp = re * (im_m * (0.5 * 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) <= -0.05)
		tmp = Float64(re * Float64(Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)) * fma(re, Float64(re * -0.16666666666666666), 1.0)));
	else
		tmp = Float64(re * Float64(im_m * Float64(0.5 * 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], -0.05], N[(re * N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(0.5 * 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.050000000000000003

   1. Initial program 65.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) + im \cdot -1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \frac{-1}{6}\right)} + im \cdot -1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6}} + im \cdot -1\right) \]

      4. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{-1 \cdot im}\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \sin re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{6} + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), \frac{-1}{6}, \mathsf{neg}\left(im\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{im \cdot \left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \color{blue}{\left(im \cdot im\right)}, \frac{-1}{6}, \mathsf{neg}\left(im\right)\right) \]

      9. neg-lowering-neg.f64 85.6

         \[\leadsto \sin re \cdot \mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, \color{blue}{-im}\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(im \cdot \left(im \cdot im\right), -0.16666666666666666, -im\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \frac{-1}{6} \cdot {im}^{3}\right) - im\right)} \]

      2. associate--l+ N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)} + \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      4. distribute-lft1-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(1 + \frac{-1}{6} \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {re}^{2} + 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      8. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{6} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto re \cdot \left(\left(\color{blue}{re \cdot \left(re \cdot \frac{-1}{6}\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto re \cdot \left(\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot re\right)} + 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto re \cdot \left(\color{blue}{\mathsf{fma}\left(re, \frac{-1}{6} \cdot re, 1\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, \color{blue}{re \cdot \frac{-1}{6}}, 1\right) \cdot \left(\frac{-1}{6} \cdot {im}^{3} - im\right)\right) \]

      15. sub-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3} + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      16. unpow3 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\frac{-1}{6} \cdot \left(\color{blue}{{im}^{2}} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      18. associate-*r* N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto re \cdot \left(\mathsf{fma}\left(re, re \cdot \frac{-1}{6}, 1\right) \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right)\right) \]

   10. Simplified 25.6%

       \[\leadsto \color{blue}{re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)} \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 72.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.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. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{60} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{60}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{60} + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. *-lowering-*.f64 89.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   5. Simplified 89.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(im \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot re\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right) \cdot re\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right) \cdot re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right)} \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      11. sub-neg N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{-2}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)} \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{re \cdot \left(\left(im \cdot \frac{1}{2}\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(im \cdot \frac{1}{2}\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right) \cdot re} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(im \cdot \frac{1}{2}\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right) \cdot re} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right)\right)} \cdot re \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right)\right)} \cdot re \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(im \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right) + -2\right)\right)}\right) \cdot re \]

      7. associate-*l* N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right)\right)} + -2\right)\right)\right) \cdot re \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right), -2\right)}\right)\right) \cdot re \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\left(im \cdot im\right) \cdot \frac{-1}{60} + \frac{-1}{3}\right)}, -2\right)\right)\right) \cdot re \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(im \cdot \left(\frac{1}{2} \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{60}, \frac{-1}{3}\right)}, -2\right)\right)\right) \cdot re \]

      11. *-lowering-*.f64 64.6

          \[\leadsto \left(im \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right) \cdot re \]

   10. Applied egg-rr 64.6%

       \[\leadsto \color{blue}{\left(im \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 52.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.05:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\\ \end{array} \end{array} \]

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) -0.05)
    (* re (* im_m (fma 0.16666666666666666 (* re re) -1.0)))
    (* re (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.05) {
		tmp = re * (im_m * fma(0.16666666666666666, (re * re), -1.0));
	} else {
		tmp = re * (im_m * 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) <= -0.05)
		tmp = Float64(re * Float64(im_m * fma(0.16666666666666666, Float64(re * re), -1.0)));
	else
		tmp = Float64(re * Float64(im_m * 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], -0.05], N[(re * N[(im$95$m * N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * 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) < -0.050000000000000003

   1. Initial program 65.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. sin-lowering-sin.f64 41.8

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Simplified 41.8%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. neg-mul-1 N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {re}^{2}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{re \cdot re}, -1\right)\right) \]

      10. *-lowering-*.f64 22.4

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{re \cdot re}, -1\right)\right) \]

   8. Simplified 22.4%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)} \]

   if -0.050000000000000003 < (sin.f64 re)

   1. Initial program 72.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(im \cdot re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

      10. *-lowering-*.f64 58.8

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

   8. Simplified 58.8%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 52.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

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 (<= im_m 3.7e-6)
    (* im_m (* re (fma im_m (* im_m -0.16666666666666666) -1.0)))
    (* re (* (* im_m (* im_m im_m)) -0.16666666666666666)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7e-6) {
		tmp = im_m * (re * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = re * ((im_m * (im_m * im_m)) * -0.16666666666666666);
	}
	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 (im_m <= 3.7e-6)
		tmp = Float64(im_m * Float64(re * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(re * Float64(Float64(im_m * Float64(im_m * im_m)) * -0.16666666666666666));
	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[im$95$m, 3.7e-6], N[(im$95$m * N[(re * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.7000000000000002e-6

   1. Initial program 59.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. *-lowering-*.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. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{60} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{60}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{60} + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. *-lowering-*.f64 93.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \]

   5. Simplified 93.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(im \cdot \color{blue}{\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot re\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right) \cdot re\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right) \cdot re} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot im\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} \cdot im\right)\right)} \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      9. *-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \frac{1}{2}\right)}\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \]

      11. sub-neg N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{-2}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}, -2\right)} \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{\left(re \cdot \left(im \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \frac{-1}{60}\right)} + \frac{-1}{3}, -2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot \frac{-1}{60}\right) \cdot im} + \frac{-1}{3}, -2\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(re \cdot \left(im \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot \frac{-1}{60}, im, \frac{-1}{3}\right)}, -2\right) \]

      4. *-lowering-*.f64 54.7

         \[\leadsto \left(re \cdot \left(im \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot -0.016666666666666666}, im, -0.3333333333333333\right), -2\right) \]

   10. Applied egg-rr 54.7%

       \[\leadsto \left(re \cdot \left(im \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im \cdot -0.016666666666666666, im, -0.3333333333333333\right)}, -2\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \color{blue}{\left(re \cdot {im}^{2}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(-1 \cdot re + \color{blue}{\left(\frac{-1}{6} \cdot re\right) \cdot {im}^{2}}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \left(\frac{-1}{6} \cdot re\right) \cdot {im}^{2}\right)} \]

       4. +-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot re\right) \cdot {im}^{2} + -1 \cdot re\right)} \]

       5. metadata-eval N/A

          \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot re\right) \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot re\right) \]

       6. cancel-sign-sub-inv N/A

          \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot re\right) \cdot {im}^{2} - 1 \cdot re\right)} \]

       7. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\frac{-1}{6} \cdot \left(re \cdot {im}^{2}\right)} - 1 \cdot re\right) \]

       8. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot re\right)} - 1 \cdot re\right) \]

       9. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re} - 1 \cdot re\right) \]

       10. distribute-rgt-out-- N/A

           \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       12. sub-neg N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto im \cdot \left(re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot im\right) + \color{blue}{-1}\right)\right) \]

       18. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot im, -1\right)}\right) \]

       19. *-commutative N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{6}}, -1\right)\right) \]

       20. *-lowering-*.f64 50.6

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.16666666666666666}, -1\right)\right) \]

   13. Simplified 50.6%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

   if 3.7000000000000002e-6 < im

   1. Initial program 99.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

      16. *-commutative N/A

          \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right) \cdot im\right)} \]

      2. flip-+ N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\frac{\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1}{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}} \cdot im\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}} \]

      4. clear-num N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sin re \cdot \frac{1}{\color{blue}{\frac{im \cdot \left(im \cdot \frac{-1}{6}\right) - -1}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

      7. sub-neg N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right) + \left(\mathsf{neg}\left(-1\right)\right)}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      8. associate-*r* N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{\left(im \cdot im\right) \cdot \frac{-1}{6}} + \left(\mathsf{neg}\left(-1\right)\right)}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      9. metadata-eval N/A

         \[\leadsto \sin re \cdot \frac{1}{\frac{\left(im \cdot im\right) \cdot \frac{-1}{6} + \color{blue}{1}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6}, 1\right)}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sin re \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}{\color{blue}{\left(\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right) - -1 \cdot -1\right) \cdot im}}} \]

   7. Applied egg-rr 31.2%

      \[\leadsto \sin re \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right) \cdot im}}} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), \frac{1}{36}, -1\right) \cdot im}} \]
   9. Step-by-step derivation

   10. Simplified 24.1%

       \[\leadsto \color{blue}{re} \cdot \frac{1}{\frac{\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)}{\mathsf{fma}\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right), 0.027777777777777776, -1\right) \cdot im}} \]

   11. Taylor expanded in im around inf

       \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{3}\right)} \]

       2. cube-mult N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) \]

       3. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto re \cdot \left(\frac{-1}{6} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       6. *-lowering-*.f64 49.1

          \[\leadsto re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   13. Simplified 49.1%

       \[\leadsto re \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 53.1% accurate, 14.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* re (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (re * (im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0)));
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(re * Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0))))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(re * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

   2. mul-1-neg N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

   3. unsub-neg N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

   4. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

   5. associate-*r* N/A

      \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

   7. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   8. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

   9. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

   11. distribute-rgt-out-- N/A

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

   12. unsub-neg N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

   14. sin-lowering-sin.f64 N/A

       \[\leadsto \color{blue}{\sin re} \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

   15. neg-mul-1 N/A

       \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{-1 \cdot im}\right) \]

   16. *-commutative N/A

       \[\leadsto \sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{im \cdot -1}\right) \]

5. Simplified 82.2%

   \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(im \cdot re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(re \cdot im\right)} \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

   6. sub-neg N/A

      \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

   10. *-lowering-*.f64 51.7

       \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

8. Simplified 51.7%

   \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
9. Add Preprocessing

Alternative 26: 33.0% accurate, 39.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(-im\_m\right) \cdot re\right) \end{array} \]

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 70.6%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

   4. sin-lowering-sin.f64 49.4

      \[\leadsto -im \cdot \color{blue}{\sin re} \]

5. Simplified 49.4%

   \[\leadsto \color{blue}{-im \cdot \sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{neg}\left(im \cdot \color{blue}{re}\right) \]
7. Step-by-step derivation

8. Simplified 33.0%

   \[\leadsto -im \cdot \color{blue}{re} \]

9. Final simplification 33.0%

   \[\leadsto \left(-im\right) \cdot re \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024199 
(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))))