math.cos on complex, imaginary part

Percentage Accurate: 65.4% → 99.7%

Time: 12.2s

Alternatives: 22

Speedup: 0.7×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.4% 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: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.5

   1. Initial program 100.0%

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

   if -0.5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 50.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

   8. Applied rewrites 87.7%

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

4. Final simplification 90.8%

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

Alternative 2: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (sin re) 0.5)) (t_1 (* t_0 (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -2e+125)
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_1 0.5)
        (*
         (*
          (fma
           (fma
            (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
            (* im_m im_m)
            -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m)
         t_0)
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = sin(re) * 0.5;
	double t_1 = t_0 * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_1 <= -2e+125) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_1 <= 0.5) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(sin(re) * 0.5)
	t_1 = Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= -2e+125)
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_1 <= 0.5)
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -2e+125], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.5], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.008333333333333333 \cdot \left(im\_m \cdot im\_m\right)\right) \cdot im\_m, im\_m, \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot \sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+125)
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 0.5)
        (*
         (*
          (fma
           (* (* -0.008333333333333333 (* im_m im_m)) im_m)
           im_m
           (fma (* -0.16666666666666666 im_m) im_m -1.0))
          (sin re))
         im_m)
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+125) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 0.5) {
		tmp = (fma(((-0.008333333333333333 * (im_m * im_m)) * im_m), im_m, fma((-0.16666666666666666 * im_m), im_m, -1.0)) * sin(re)) * im_m;
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+125)
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.5)
		tmp = Float64(Float64(fma(Float64(Float64(-0.008333333333333333 * Float64(im_m * im_m)) * im_m), im_m, fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * sin(re)) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+125], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.6%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right)\right) \cdot im, im, \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot \sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+125)
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 0.5)
        (*
         (*
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0)
          (sin re))
         im_m)
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+125) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 0.5) {
		tmp = (fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0) * sin(re)) * im_m;
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+125)
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.5)
		tmp = Float64(Float64(fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0) * sin(re)) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+125], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right) \cdot \sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot \left(\sin re \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+125)
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 0.5)
        (* (fma (* -0.16666666666666666 im_m) im_m -1.0) (* (sin re) im_m))
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+125) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 0.5) {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * (sin(re) * im_m);
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+125)
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.5)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * Float64(sin(re) * im_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+125], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

   8. Applied rewrites 99.4%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot \left(\sin re \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\sin re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+125)
      (* (* re 0.5) (- 1.0 (exp im_m)))
      (if (<= t_0 0.5)
        (* (sin re) (- im_m))
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+125) {
		tmp = (re * 0.5) * (1.0 - exp(im_m));
	} else if (t_0 <= 0.5) {
		tmp = sin(re) * -im_m;
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+125)
		tmp = Float64(Float64(re * 0.5) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.5)
		tmp = Float64(sin(re) * Float64(-im_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+125], N[(N[(re * 0.5), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[Sin[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 39.0

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

   8. Applied rewrites 39.0%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\sin re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+125)
      (*
       (*
        (fma
         (fma 0.004166666666666667 (* re re) -0.08333333333333333)
         (* re re)
         0.5)
        re)
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m))
      (if (<= t_0 0.5)
        (* (sin re) (- im_m))
        (*
         (*
          (* (* (* re im_m) re) -0.16666666666666666)
          (fma
           (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
           (* im_m im_m)
           -1.0))
         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 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+125) {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	} else if (t_0 <= 0.5) {
		tmp = sin(re) * -im_m;
	} else {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * 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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+125)
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	elseif (t_0 <= 0.5)
		tmp = Float64(sin(re) * Float64(-im_m));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * 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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+125], N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[Sin[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e125

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   if -1.9999999999999998e125 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.5

   1. Initial program 28.6%

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

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.5:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+231}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), {im\_m}^{4}, \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot \sin re\right) \cdot im\_m\\ \end{array} \end{array} \]

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)) -2e+231)
    (* (- 1.0 (exp im_m)) (* (sin re) 0.5))
    (*
     (*
      (fma
       (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
       (pow im_m 4.0)
       (fma (* -0.16666666666666666 im_m) im_m -1.0))
      (sin re))
     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)) <= -2e+231) {
		tmp = (1.0 - exp(im_m)) * (sin(re) * 0.5);
	} else {
		tmp = (fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), pow(im_m, 4.0), fma((-0.16666666666666666 * im_m), im_m, -1.0)) * sin(re)) * 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(exp(Float64(-im_m)) - exp(im_m)) <= -2e+231)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(sin(re) * 0.5));
	else
		tmp = Float64(Float64(fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (im_m ^ 4.0), fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * sin(re)) * 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[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+231], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[Power[im$95$m, 4.0], $MachinePrecision] + N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2.0000000000000001e231

   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

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

   5. Applied rewrites 100.0%

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

   if -2.0000000000000001e231 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+231}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), {im}^{4}, \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot \sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+231}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (sin re) 0.5)))
   (*
    im_s
    (if (<= (- (exp (- im_m)) (exp im_m)) -2e+231)
      (* (- 1.0 (exp im_m)) t_0)
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = sin(re) * 0.5;
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+231) {
		tmp = (1.0 - exp(im_m)) * t_0;
	} else {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(sin(re) * 0.5)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+231)
		tmp = Float64(Float64(1.0 - exp(im_m)) * t_0);
	else
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+231], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2.0000000000000001e231

   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

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

   5. Applied rewrites 100.0%

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

   if -2.0000000000000001e231 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+231}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.01)
    (*
     (*
      (* (* (* re im_m) re) -0.16666666666666666)
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0))
     re)
    (*
     (*
      (fma
       (fma 0.004166666666666667 (* re re) -0.08333333333333333)
       (* re re)
       0.5)
      re)
     (*
      (fma
       (fma
        (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
        (* im_m im_m)
        -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * re;
	} else {
		tmp = (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re) * (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * re);
	else
		tmp = Float64(Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 31.0%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 29.5%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.01)
    (*
     (*
      (* (* (* re im_m) re) -0.16666666666666666)
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0))
     re)
    (*
     (*
      (fma
       (fma
        (* -0.0003968253968253968 (* im_m im_m))
        (* im_m im_m)
        -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)
     (*
      (fma
       (fma 0.004166666666666667 (* re re) -0.08333333333333333)
       (* re re)
       0.5)
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * re;
	} else {
		tmp = (fma(fma((-0.0003968253968253968 * (im_m * im_m)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (fma(fma(0.004166666666666667, (re * re), -0.08333333333333333), (re * re), 0.5) * re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * re);
	else
		tmp = Float64(Float64(fma(fma(Float64(-0.0003968253968253968 * Float64(im_m * im_m)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(fma(fma(0.004166666666666667, Float64(re * re), -0.08333333333333333), Float64(re * re), 0.5) * re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 31.0%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 29.5%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 68.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.004166666666666667, re \cdot re, -0.08333333333333333\right), re \cdot re, 0.5\right) \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (*
     (*
      (* (* (* re im_m) re) -0.16666666666666666)
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0))
     re)
    (*
     (* re 0.5)
     (*
      (fma
       (fma
        (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
        (* im_m im_m)
        -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * re;
	} else {
		tmp = (re * 0.5) * (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * re);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 31.0%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 29.5%

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

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, re \cdot re, im\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, re \cdot re, 0.16666666666666666\right) \cdot im\_m, re \cdot re, -im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) 2e-6)
    (*
     (*
      (fma (* -0.16666666666666666 im_m) (* re re) im_m)
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0))
     re)
    (*
     (fma
      (* (fma -0.008333333333333333 (* re re) 0.16666666666666666) im_m)
      (* re re)
      (- im_m))
     re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= 2e-6) {
		tmp = (fma((-0.16666666666666666 * im_m), (re * re), im_m) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * re;
	} else {
		tmp = fma((fma(-0.008333333333333333, (re * re), 0.16666666666666666) * im_m), (re * re), -im_m) * re;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= 2e-6)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * im_m), Float64(re * re), im_m) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * re);
	else
		tmp = Float64(fma(Float64(fma(-0.008333333333333333, Float64(re * re), 0.16666666666666666) * im_m), Float64(re * re), Float64(-im_m)) * re);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 2e-6], N[(N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + im$95$m), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.99999999999999991e-6

   1. Initial program 68.5%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 71.5%

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

   if 1.99999999999999991e-6 < (sin.f64 re)

   1. Initial program 47.3%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 21.0%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, re \cdot re, im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, re \cdot re, 0.16666666666666666\right) \cdot im, re \cdot re, -im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

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) -0.02)
    (*
     (*
      (* (* (* re im_m) re) -0.16666666666666666)
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0))
     re)
    (*
     (* (fma (* -0.016666666666666666 (* im_m im_m)) (* im_m im_m) -2.0) im_m)
     (* re 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = ((((re * im_m) * re) * -0.16666666666666666) * fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0)) * re;
	} else {
		tmp = (fma((-0.016666666666666666 * (im_m * im_m)), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(Float64(Float64(Float64(re * im_m) * re) * -0.16666666666666666) * fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0)) * re);
	else
		tmp = Float64(Float64(fma(Float64(-0.016666666666666666 * Float64(im_m * im_m)), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 31.0%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 29.5%

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

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.7

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 65.7%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

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) -0.02)
    (*
     (* (fma -0.3333333333333333 (* im_m im_m) -2.0) im_m)
     (* (fma (* re re) -0.08333333333333333 0.5) re))
    (*
     (* (fma (* -0.016666666666666666 (* im_m im_m)) (* im_m im_m) -2.0) im_m)
     (* re 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (fma(-0.3333333333333333, (im_m * im_m), -2.0) * im_m) * (fma((re * re), -0.08333333333333333, 0.5) * re);
	} else {
		tmp = (fma((-0.016666666666666666 * (im_m * im_m)), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * im_m) * Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re));
	else
		tmp = Float64(Float64(fma(Float64(-0.016666666666666666 * Float64(im_m * im_m)), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.0

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

   8. Applied rewrites 31.0%

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

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.7

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 65.7%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\_m\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

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) -0.02)
    (* (* 0.16666666666666666 (* (* re im_m) re)) re)
    (*
     (* (fma (* -0.016666666666666666 (* im_m im_m)) (* im_m im_m) -2.0) im_m)
     (* re 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = (fma((-0.016666666666666666 * (im_m * im_m)), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(0.16666666666666666 * Float64(Float64(re * im_m) * re)) * re);
	else
		tmp = Float64(Float64(fma(Float64(-0.016666666666666666 * Float64(im_m * im_m)), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(0.16666666666666666 * N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 28.0%

       \[\leadsto \left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.7

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 65.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\_m\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im\_m, im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (* 0.16666666666666666 (* (* re im_m) re)) re)
    (*
     (*
      (fma
       (fma (* -0.008333333333333333 im_m) im_m -0.16666666666666666)
       (* im_m im_m)
       -1.0)
      im_m)
     re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = (fma(fma((-0.008333333333333333 * im_m), im_m, -0.16666666666666666), (im_m * im_m), -1.0) * im_m) * re;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(0.16666666666666666 * Float64(Float64(re * im_m) * re)) * re);
	else
		tmp = Float64(Float64(fma(fma(Float64(-0.008333333333333333 * im_m), im_m, -0.16666666666666666), Float64(im_m * im_m), -1.0) * im_m) * re);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(0.16666666666666666 * N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(-0.008333333333333333 * im$95$m), $MachinePrecision] * im$95$m + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 28.0%

       \[\leadsto \left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 65.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333 \cdot im, im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\_m\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

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) -0.02)
    (* (* 0.16666666666666666 (* (* re im_m) re)) re)
    (* (* (fma -0.3333333333333333 (* im_m im_m) -2.0) im_m) (* re 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = (fma(-0.3333333333333333, (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(0.16666666666666666 * Float64(Float64(re * im_m) * re)) * re);
	else
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(0.16666666666666666 * N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 28.0%

       \[\leadsto \left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.2

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

   8. Applied rewrites 60.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\_m\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

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) -0.02)
    (* (* 0.16666666666666666 (* (* re im_m) re)) re)
    (* (* (fma (* -0.16666666666666666 im_m) im_m -1.0) re) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = (fma((-0.16666666666666666 * im_m), im_m, -1.0) * re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(0.16666666666666666 * Float64(Float64(re * im_m) * re)) * re);
	else
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(0.16666666666666666 * N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 28.0%

       \[\leadsto \left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 63.7%

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

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 55.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 68.5%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.3%

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

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

   1. Initial program 47.3%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 10.4%

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

4. Final simplification 34.9%

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

Alternative 21: 34.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\_m\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (* 0.16666666666666666 (* (* re im_m) re)) re)
    (* re (- im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = re * -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (sin(re) <= (-0.02d0)) then
        tmp = (0.16666666666666666d0 * ((re * im_m) * re)) * re
    else
        tmp = re * -im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (Math.sin(re) <= -0.02) {
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	} else {
		tmp = re * -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if math.sin(re) <= -0.02:
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re
	else:
		tmp = re * -im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(Float64(0.16666666666666666 * Float64(Float64(re * im_m) * re)) * re);
	else
		tmp = Float64(re * Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (sin(re) <= -0.02)
		tmp = (0.16666666666666666 * ((re * im_m) * re)) * re;
	else
		tmp = re * -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(0.16666666666666666 * N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision], N[(re * (-im$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 28.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 28.0%

       \[\leadsto \left(\left(\left(im \cdot re\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 64.6%

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

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sin.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.2%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\left(0.16666666666666666 \cdot \left(\left(re \cdot im\right) \cdot re\right)\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 32.5% accurate, 39.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot \left(-im\_m\right)\right) \end{array} \]

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))))

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);
}

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 * (re * -im_m)
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 * (re * -im_m);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (re * -im_m)

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)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (re * -im_m);
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 * (-im$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

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

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

   2. lower-*.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 31.0%

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

9. Final simplification 31.0%

   \[\leadsto re \cdot \left(-im\right) \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024255 
(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))))