math.cos on complex, imaginary part

Percentage Accurate: 66.6% → 99.5%

Time: 15.0s

Alternatives: 20

Speedup: 0.7×

• 50.3% 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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

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))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (fma
         (* im_m im_m)
         (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
         -2.0)))))))

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * fma(Float64(im_m * im_m), fma(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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 59.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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 92.9

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

   5. Simplified 92.9%

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

4. Final simplification 94.7%

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

Alternative 2: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* (- (exp (- im_m)) (exp im_m)) t_0)))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (if (<= t_1 2e-7)
        (*
         t_0
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
           -2.0)))
        (*
         im_m
         (*
          (fma -0.16666666666666666 (* re (* re re)) re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double t_1 = (exp(-im_m) - exp(im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else if (t_1 <= 2e-7) {
		tmp = t_0 * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0));
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	elseif (t_1 <= 2e-7)
		tmp = Float64(t_0 * 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(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.6%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 32.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

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

   5. Simplified 99.8%

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

   if 1.9999999999999999e-7 < (*.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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 79.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 67.6%

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

4. Final simplification 73.0%

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

Alternative 3: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ t_1 := \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))))
        (t_1
         (fma
          (* im_m im_m)
          (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
          -1.0)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (if (<= t_0 2e-7)
        (* im_m (* (sin re) t_1))
        (* im_m (* (fma -0.16666666666666666 (* re (* re re)) re) t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double t_1 = fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else if (t_0 <= 2e-7) {
		tmp = im_m * (sin(re) * t_1);
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * t_1);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	t_1 = fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 2e-7)
		tmp = Float64(im_m * Float64(sin(re) * t_1));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * t_1));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.6%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 32.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. *-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 99.8%

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

   if 1.9999999999999999e-7 < (*.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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 79.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 67.6%

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

4. Final simplification 73.0%

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

Alternative 4: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (if (<= t_0 2e-7)
        (* (sin re) (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         im_m
         (*
          (fma -0.16666666666666666 (* re (* re re)) re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else if (t_0 <= 2e-7) {
		tmp = sin(re) * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 2e-7)
		tmp = Float64(sin(re) * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.6%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 32.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   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 99.8%

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

   if 1.9999999999999999e-7 < (*.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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 79.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 67.6%

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

4. Final simplification 73.0%

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

Alternative 5: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-im\_m \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (if (<= t_0 2e-7)
        (- (* im_m (sin re)))
        (*
         im_m
         (*
          (fma -0.16666666666666666 (* re (* re re)) re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else if (t_0 <= 2e-7) {
		tmp = -(im_m * sin(re));
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 2e-7)
		tmp = Float64(-Float64(im_m * sin(re)));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.6%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 32.7%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 99.0

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

   5. Simplified 99.0%

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

   if 1.9999999999999999e-7 < (*.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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 79.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 67.6%

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

4. Final simplification 72.7%

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

Alternative 6: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(re \cdot \left(re \cdot 0.008333333333333333\right), t\_0, re\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-im\_m \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, t\_0, re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* re (* re re)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (*
       (*
        im_m
        (fma
         (* im_m im_m)
         (fma
          (* im_m im_m)
          (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
          -0.3333333333333333)
         -2.0))
       (* 0.5 (fma (* re (* re 0.008333333333333333)) t_0 re)))
      (if (<= t_1 2e-7)
        (- (* im_m (sin re)))
        (*
         im_m
         (*
          (fma -0.16666666666666666 t_0 re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = re * (re * re);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (0.5 * fma((re * (re * 0.008333333333333333)), t_0, re));
	} else if (t_1 <= 2e-7) {
		tmp = -(im_m * sin(re));
	} else {
		tmp = im_m * (fma(-0.16666666666666666, t_0, re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(re * Float64(re * re))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(0.5 * fma(Float64(re * Float64(re * 0.008333333333333333)), t_0, re)));
	elseif (t_1 <= 2e-7)
		tmp = Float64(-Float64(im_m * sin(re)));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, t_0, re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + \color{blue}{re}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot re\right) + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow3 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{{re}^{2}}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      19. *-lowering-*.f64 73.2

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Simplified 73.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2}}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{120}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot \frac{1}{120}\right)}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot \frac{1}{120}\right)}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. *-lowering-*.f64 73.2

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(re \cdot \color{blue}{\left(re \cdot 0.008333333333333333\right)}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Simplified 73.2%

       \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\color{blue}{re \cdot \left(re \cdot 0.008333333333333333\right)}, re \cdot \left(re \cdot re\right), re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 32.5%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 99.0

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

   5. Simplified 99.0%

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

   if 1.9999999999999999e-7 < (*.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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 79.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 67.6%

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

4. Final simplification 83.3%

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

Alternative 7: 90.3% accurate, 0.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (* (- (exp (- im_m)) (exp im_m)) t_0) (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (*
       t_0
       (*
        im_m
        (fma
         (* im_m im_m)
         (fma
          (* im_m im_m)
          (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
          -0.3333333333333333)
         -2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * t_0) <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = t_0 * (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

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

Wolfram

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

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 74.6%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 32.7%

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

   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 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

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 78.6%

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

Alternative 8: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -4e-15)
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
        -0.3333333333333333)
       -2.0)))
    (*
     im_m
     (*
      (fma -0.16666666666666666 (* re (* re re)) re)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -4e-15) {
		tmp = (0.5 * re) * (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0));
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -4e-15)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-15], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.0000000000000003e-15

   1. Initial program 98.4%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 95.8

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

   5. Simplified 95.8%

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

   6. Taylor expanded in re around 0

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

      1. *-lowering-*.f64 67.6

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

   8. Simplified 67.6%

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

   if -4.0000000000000003e-15 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.5%

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

   3. Taylor expanded in im around 0

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

      1. *-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 91.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 57.2

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

   8. Simplified 57.2%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 58.9%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -4e-15)
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
       -2.0))
     (* 0.5 re))
    (*
     im_m
     (*
      (fma -0.16666666666666666 (* re (* re re)) re)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -4e-15) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * (0.5 * re);
	} else {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -4e-15)
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * Float64(0.5 * re));
	else
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-15], N[(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] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -4.0000000000000003e-15

   1. Initial program 98.4%

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

   3. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 72.9

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

   5. Simplified 72.9%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 71.7%

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

   9. Taylor expanded in im around 0

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 63.5

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

   11. Simplified 63.5%

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

   if -4.0000000000000003e-15 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.5%

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

   3. Taylor expanded in im around 0

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

      1. *-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 91.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 57.2

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

   8. Simplified 57.2%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 58.9%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -1e-45)
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
       -2.0))
     (* 0.5 re))
    (*
     (* re (fma re (* re -0.08333333333333333) 0.5))
     (* im_m (fma im_m (* im_m -0.3333333333333333) -2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -1e-45) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * (0.5 * re);
	} else {
		tmp = (re * fma(re, (re * -0.08333333333333333), 0.5)) * (im_m * fma(im_m, (im_m * -0.3333333333333333), -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -1e-45)
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)) * Float64(im_m * fma(im_m, Float64(im_m * -0.3333333333333333), -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-45], N[(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] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -9.99999999999999984e-46

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 72.0

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

   5. Simplified 72.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 70.7%

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

   9. Taylor expanded in im around 0

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 62.7

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

   11. Simplified 62.7%

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

   if -9.99999999999999984e-46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.7%

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

   3. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 49.4

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

   5. Simplified 49.4%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-lowering-*.f64 56.5

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

   8. Simplified 56.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im\_m \cdot 0.16666666666666666\right), -im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -1e-45)
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.016666666666666666 -0.3333333333333333)
       -2.0))
     (* 0.5 re))
    (* re (fma re (* re (* im_m 0.16666666666666666)) (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -1e-45) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * (0.5 * re);
	} else {
		tmp = re * fma(re, (re * (im_m * 0.16666666666666666)), -im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -1e-45)
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.016666666666666666, -0.3333333333333333), -2.0)) * Float64(0.5 * re));
	else
		tmp = Float64(re * fma(re, Float64(re * Float64(im_m * 0.16666666666666666)), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-45], N[(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] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $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))) < -9.99999999999999984e-46

   1. Initial program 97.4%

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

   3. Taylor expanded in re around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 72.0

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

   5. Simplified 72.0%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 70.7%

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

   9. Taylor expanded in im around 0

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. metadata-eval N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 62.7

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

   11. Simplified 62.7%

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

   if -9.99999999999999984e-46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. cancel-sign-sub-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. neg-lowering-neg.f64 39.6

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

   8. Simplified 39.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot 0.16666666666666666\right), -im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im\_m \cdot 0.16666666666666666\right), -im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -1e-45)
    (*
     im_m
     (*
      re
      (fma
       im_m
       (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
       -1.0)))
    (* re (fma re (* re (* im_m 0.16666666666666666)) (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -1e-45) {
		tmp = im_m * (re * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0));
	} else {
		tmp = re * fma(re, (re * (im_m * 0.16666666666666666)), -im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -1e-45)
		tmp = Float64(im_m * Float64(re * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0)));
	else
		tmp = Float64(re * fma(re, Float64(re * Float64(im_m * 0.16666666666666666)), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-45], N[(im$95$m * N[(re * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $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))) < -9.99999999999999984e-46

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 85.3%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 60.0

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

   8. Simplified 60.0%

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

   if -9.99999999999999984e-46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. cancel-sign-sub-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. neg-lowering-neg.f64 39.6

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

   8. Simplified 39.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot 0.16666666666666666\right), -im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im\_m \cdot 0.16666666666666666\right), -im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -1e-45)
    (* re (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0)))
    (* re (fma re (* re (* im_m 0.16666666666666666)) (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -1e-45) {
		tmp = re * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = re * fma(re, (re * (im_m * 0.16666666666666666)), -im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -1e-45)
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(re * fma(re, Float64(re * Float64(im_m * 0.16666666666666666)), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-45], N[(re * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $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))) < -9.99999999999999984e-46

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

      13. *-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 70.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 54.4%

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

   if -9.99999999999999984e-46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. cancel-sign-sub-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. neg-lowering-neg.f64 39.6

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

   8. Simplified 39.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot 0.16666666666666666\right), -im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(re, re \cdot 0.16666666666666666, -1\right)\right)\\ \end{array} \end{array} \]

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)) (* 0.5 (sin re))) -1e-45)
    (* re (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0)))
    (* re (* im_m (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)) * (0.5 * sin(re))) <= -1e-45) {
		tmp = re * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = re * (im_m * 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(0.5 * sin(re))) <= -1e-45)
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(re * Float64(im_m * 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-45], N[(re * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * 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))) < -9.99999999999999984e-46

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-*r* N/A

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

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

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

      12. unsub-neg N/A

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

      13. *-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 70.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 54.4%

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

   if -9.99999999999999984e-46 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. 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 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in re around 0

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

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

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. cancel-sign-sub-inv N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-out N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. neg-lowering-neg.f64 39.6

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

   8. Simplified 39.6%

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

   9. Taylor expanded in re around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-rgt-out N/A

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

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

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

       11. *-lowering-*.f64 40.7

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

   11. Simplified 40.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -1 \cdot 10^{-45}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(re, re \cdot 0.16666666666666666, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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)) * (0.5 * Math.sin(re))) <= -Double.POSITIVE_INFINITY) {
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)));
	} 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)) * (0.5 * math.sin(re))) <= -math.inf:
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)))
	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(0.5 * sin(re))) <= Float64(-Inf))
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(re * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * Float64(-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)) * (0.5 * sin(re))) <= -Inf)
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)));
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(im$95$m * N[(-0.16666666666666666 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $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))) < -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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   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 69.0%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 46.0

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

   8. Simplified 46.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

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

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 46.0

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

   11. Simplified 46.0%

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

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

   1. Initial program 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

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   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 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. neg-lowering-neg.f64 36.6

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

   8. Simplified 36.6%

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

4. Final simplification 39.1%

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

Alternative 16: 60.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := re \cdot \left(re \cdot re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, t\_0, re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), t\_0, re\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* re (* re re))))
   (*
    im_s
    (if (<= (sin re) -0.01)
      (*
       im_m
       (*
        (fma -0.16666666666666666 t_0 re)
        (fma
         (* im_m im_m)
         (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
         -1.0)))
      (*
       (*
        im_m
        (fma
         (* im_m im_m)
         (fma
          (* im_m im_m)
          (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
          -0.3333333333333333)
         -2.0))
       (*
        0.5
        (fma
         (fma (* re re) 0.008333333333333333 -0.16666666666666666)
         t_0
         re)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = re * (re * re);
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = im_m * (fma(-0.16666666666666666, t_0, re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (0.5 * fma(fma((re * re), 0.008333333333333333, -0.16666666666666666), t_0, re));
	}
	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(re * Float64(re * re))
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, t_0, re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(0.5 * fma(fma(Float64(re * re), 0.008333333333333333, -0.16666666666666666), t_0, 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_] := Block[{t$95$0 = N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(im$95$m * N[(N[(-0.16666666666666666 * t$95$0 + 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[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * t$95$0 + re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

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

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

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

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

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

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 24.6

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

   8. Simplified 24.6%

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

   9. Taylor expanded in re around 0

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

   11. Simplified 28.5%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 74.8%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + 1 \cdot re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re + \color{blue}{re}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2}\right)} \cdot re + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left({re}^{2} \cdot re\right)} + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot re\right) + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow3 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

         \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{{re}^{2}}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

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

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      19. *-lowering-*.f64 79.1

          \[\leadsto \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Simplified 79.1%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.01)
    (*
     im_m
     (*
      (fma -0.16666666666666666 (* re (* re re)) re)
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0)))
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
        -0.3333333333333333)
       -2.0))
     (*
      re
      (fma
       (* re re)
       (fma (* re re) 0.004166666666666667 -0.08333333333333333)
       0.5))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = im_m * (fma(-0.16666666666666666, (re * (re * re)), re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), fma((re * re), 0.004166666666666667, -0.08333333333333333), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(im_m * Float64(fma(-0.16666666666666666, Float64(re * Float64(re * re)), re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), fma(Float64(re * re), 0.004166666666666667, -0.08333333333333333), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(im$95$m * N[(N[(-0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * 0.004166666666666667 + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot re\right) + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      7. unpow3 N/A

         \[\leadsto im \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot \color{blue}{{re}^{3}} + re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{3}, re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      9. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      10. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{120}, \frac{-1}{6}\right), {re}^{3}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      15. cube-mult N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot \left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      16. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{{re}^{2}}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), \color{blue}{re \cdot {re}^{2}}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      18. unpow2 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{-1}{6}\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      19. *-lowering-*.f64 24.6

          \[\leadsto im \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \color{blue}{\left(re \cdot re\right)}, re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   8. Simplified 24.6%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)} \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6}}, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 28.5%

       \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{-0.16666666666666666}, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 74.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. *-lowering-*.f64 95.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Simplified 95.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{240} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{240}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, {re}^{2} \cdot \frac{1}{240} + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{240}, \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{240}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. *-lowering-*.f64 79.1

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(\color{blue}{re \cdot re}, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Simplified 79.1%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot \left(re \cdot re\right), re\right) \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, 0.004166666666666667, -0.08333333333333333\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right) \end{array} \]

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 im_m (* im_m -0.16666666666666666) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (re * (im_m * fma(im_m, (im_m * -0.16666666666666666), -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(im_m, Float64(im_m * -0.16666666666666666), -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[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

   2. mul-1-neg N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

   3. unsub-neg N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

   4. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

   5. associate-*r* N/A

      \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

   7. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   8. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

   9. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

   11. distribute-rgt-out-- N/A

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

   12. unsub-neg N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

   13. *-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 81.6%

   \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right) \]
7. Step-by-step derivation

8. Simplified 53.9%

   \[\leadsto \color{blue}{re} \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \]
9. Add Preprocessing

Alternative 19: 51.8% accurate, 14.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right) \end{array} \]

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 (* im_m (* re (fma im_m (* im_m -0.16666666666666666) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * fma(im_m, (im_m * -0.16666666666666666), -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(im_m * Float64(re * fma(im_m, Float64(im_m * -0.16666666666666666), -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[(im$95$m * N[(re * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.4%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

   2. mul-1-neg N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)}\right) \]

   3. unsub-neg N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

   4. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

   5. associate-*r* N/A

      \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

   7. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   8. *-commutative N/A

      \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

   9. associate-*r* N/A

      \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

   11. distribute-rgt-out-- N/A

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

   12. unsub-neg N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

   13. *-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 81.6%

   \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]

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. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot im\right) \cdot im} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot im\right)} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto im \cdot \left(re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot im\right) + \color{blue}{-1}\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(im, \frac{-1}{6} \cdot im, -1\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{6}}, -1\right)\right) \]

   10. *-lowering-*.f64 49.0

       \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.16666666666666666}, -1\right)\right) \]

8. Simplified 49.0%

   \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)} \]
9. Add Preprocessing

Alternative 20: 34.3% accurate, 39.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-re\right)\right) \end{array} \]

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 (* 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(im_m * Float64(-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 69.4%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   2. 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 47.2

      \[\leadsto -im \cdot \color{blue}{\sin re} \]

5. Simplified 47.2%

   \[\leadsto \color{blue}{-im \cdot \sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot re} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(im \cdot -1\right)} \cdot re \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re\right)} \]

   5. neg-mul-1 N/A

      \[\leadsto im \cdot \color{blue}{\left(\mathsf{neg}\left(re\right)\right)} \]

   6. neg-lowering-neg.f64 30.4

      \[\leadsto im \cdot \color{blue}{\left(-re\right)} \]

8. Simplified 30.4%

   \[\leadsto \color{blue}{im \cdot \left(-re\right)} \]
9. 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 2024204 
(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))))