math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.6%

Time: 15.1s

Alternatives: 28

Speedup: 0.7×

• 49.5% 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: 65.7% 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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 47.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 95.3%

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

Alternative 2: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im\_m, im\_m \cdot -0.0001984126984126984, -0.008333333333333333\right), im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* t_0 (* 0.5 re))
      (if (<= t_1 4e-6)
        (*
         im_m
         (*
          (sin re)
          (fma
           (fma im_m (* im_m -0.0001984126984126984) -0.008333333333333333)
           (* im_m (* im_m (* im_m im_m)))
           (fma -0.16666666666666666 (* im_m im_m) -1.0))))
        (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * re);
	} else if (t_1 <= 4e-6) {
		tmp = im_m * (sin(re) * fma(fma(im_m, (im_m * -0.0001984126984126984), -0.008333333333333333), (im_m * (im_m * (im_m * im_m))), fma(-0.16666666666666666, (im_m * im_m), -1.0)));
	} else {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (t_1 <= 4e-6)
		tmp = Float64(im_m * Float64(sin(re) * fma(fma(im_m, Float64(im_m * -0.0001984126984126984), -0.008333333333333333), Float64(im_m * Float64(im_m * Float64(im_m * im_m))), fma(-0.16666666666666666, Float64(im_m * im_m), -1.0))));
	else
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-6], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im, im \cdot -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ t_2 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot t\_1\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* (- (exp (- im_m)) (exp im_m)) t_1)))
   (*
    im_s
    (if (<= t_2 (- INFINITY))
      (* t_0 (* 0.5 re))
      (if (<= t_2 4e-6)
        (*
         t_1
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0)))
        (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double t_2 = (exp(-im_m) - exp(im_m)) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * re);
	} else if (t_2 <= 4e-6) {
		tmp = t_1 * (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0));
	} else {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (t_2 <= 4e-6)
		tmp = Float64(t_1 * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-6], N[(t$95$1 * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* t_0 (* 0.5 re))
      (if (<= t_1 4e-6)
        (*
         im_m
         (*
          (sin re)
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
           -1.0)))
        (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * re);
	} else if (t_1 <= 4e-6) {
		tmp = im_m * (sin(re) * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	} else {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (t_1 <= 4e-6)
		tmp = Float64(im_m * Float64(sin(re) * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	else
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-6], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 1 - e^{im\_m}\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (exp im_m)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* t_0 (* 0.5 re))
      (if (<= t_1 4e-6)
        (* im_m (* (sin re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (* t_0 (* re (fma re (* re -0.08333333333333333) 0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 1.0 - exp(im_m);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * re);
	} else if (t_1 <= 4e-6) {
		tmp = im_m * (sin(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = t_0 * (re * fma(re, (re * -0.08333333333333333), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(1.0 - exp(im_m))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * re));
	elseif (t_1 <= 4e-6)
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(t_0 * Float64(re * fma(re, Float64(re * -0.08333333333333333), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-6], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(re \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= t_0 4e-6)
        (* im_m (* (sin re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0))
         (*
          re
          (fma
           (* re re)
           (* re (* re (* (* re re) -9.92063492063492e-5)))
           0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (t_0 <= 4e-6) {
		tmp = im_m * (sin(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (t_0 <= 4e-6)
		tmp = Float64(im_m * Float64(sin(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-6], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 58.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 58.4

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= t_0 4e-6)
        (* (* im_m (sin re)) (fma (* im_m im_m) -0.16666666666666666 -1.0))
        (*
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0))
         (*
          re
          (fma
           (* re re)
           (* re (* re (* (* re re) -9.92063492063492e-5)))
           0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (t_0 <= 4e-6) {
		tmp = (im_m * sin(re)) * fma((im_m * im_m), -0.16666666666666666, -1.0);
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (t_0 <= 4e-6)
		tmp = Float64(Float64(im_m * sin(re)) * fma(Float64(im_m * im_m), -0.16666666666666666, -1.0));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-6], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. mul-1-neg N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 58.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 58.4

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sin re \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= t_0 4e-6)
        (* (sin re) (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0)))
        (*
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0))
         (*
          re
          (fma
           (* re re)
           (* re (* re (* (* re re) -9.92063492063492e-5)))
           0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (t_0 <= 4e-6) {
		tmp = sin(re) * (im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0));
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (t_0 <= 4e-6)
		tmp = Float64(sin(re) * Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-6], N[(N[Sin[re], $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 58.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 58.4

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= t_0 4e-6)
        (* (- im_m) (sin re))
        (*
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0))
         (*
          re
          (fma
           (* re re)
           (* re (* re (* (* re re) -9.92063492063492e-5)))
           0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (t_0 <= 4e-6) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (t_0 <= 4e-6)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-6], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.1%

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

   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. lower-*.f64 45.2

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

   8. Applied rewrites 45.2%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 58.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 58.4

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\\ t_1 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma
            (* im_m im_m)
            (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
            -0.3333333333333333)
           -2.0)))
        (t_1 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re)))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (* 0.5 re) t_0)
      (if (<= t_1 4e-6)
        (* (- im_m) (sin re))
        (*
         t_0
         (*
          re
          (fma
           (* re re)
           (* re (* re (* (* re re) -9.92063492063492e-5)))
           0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0);
	double t_1 = (exp(-im_m) - exp(im_m)) * (0.5 * sin(re));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * t_0;
	} else if (t_1 <= 4e-6) {
		tmp = -im_m * sin(re);
	} else {
		tmp = t_0 * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0))
	t_1 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * t_0);
	elseif (t_1 <= 4e-6)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(t_0 * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 4e-6], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 65.8

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

   8. Applied rewrites 65.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 3.99999999999999982e-6

   1. Initial program 30.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 3.99999999999999982e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 99.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 58.4

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 58.4

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 58.4%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) 0.0)
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
        -0.3333333333333333)
       -2.0)))
    (*
     (- im_m)
     (fma
      (fma
       (* re re)
       (fma (* re re) -0.0001984126984126984 0.008333333333333333)
       -0.16666666666666666)
      (* re (* re re))
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= 0.0) {
		tmp = (0.5 * re) * (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0));
	} else {
		tmp = -im_m * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= 0.0)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(Float64(-im_m) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 58.0

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

   8. Applied rewrites 58.0%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 9.7

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

   5. Applied rewrites 9.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 23.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) 0.0)
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (* (* im_m im_m) -0.0003968253968253968)
        -0.3333333333333333)
       -2.0)))
    (*
     (- im_m)
     (fma
      (fma
       (* re re)
       (fma (* re re) -0.0001984126984126984 0.008333333333333333)
       -0.16666666666666666)
      (* re (* re re))
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= 0.0) {
		tmp = (0.5 * re) * (im_m * fma((im_m * im_m), fma((im_m * im_m), ((im_m * im_m) * -0.0003968253968253968), -0.3333333333333333), -2.0));
	} else {
		tmp = -im_m * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= 0.0)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * -0.0003968253968253968), -0.3333333333333333), -2.0)));
	else
		tmp = Float64(Float64(-im_m) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 58.0

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

   8. Applied rewrites 58.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 58.0

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

   11. Applied rewrites 58.0%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 9.7

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

   5. Applied rewrites 9.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 23.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq 0:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot -0.0003968253968253968, -0.3333333333333333\right), -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re \cdot re, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re \cdot \left(re \cdot re\right), re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.5% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) 0.0)
    (*
     (* 0.5 re)
     (*
      im_m
      (fma
       (* im_m im_m)
       (* (* im_m (* im_m im_m)) (* im_m -0.0003968253968253968))
       -2.0)))
    (*
     (- im_m)
     (fma
      (fma
       (* re re)
       (fma (* re re) -0.0001984126984126984 0.008333333333333333)
       -0.16666666666666666)
      (* re (* re re))
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= 0.0) {
		tmp = (0.5 * re) * (im_m * fma((im_m * im_m), ((im_m * (im_m * im_m)) * (im_m * -0.0003968253968253968)), -2.0));
	} else {
		tmp = -im_m * fma(fma((re * re), fma((re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (re * (re * re)), re);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= 0.0)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * fma(Float64(im_m * im_m), Float64(Float64(im_m * Float64(im_m * im_m)) * Float64(im_m * -0.0003968253968253968)), -2.0)));
	else
		tmp = Float64(Float64(-im_m) * fma(fma(Float64(re * re), fma(Float64(re * re), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(re * Float64(re * re)), re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(N[(N[(re * re), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 58.0

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

   8. Applied rewrites 58.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 57.9

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

   11. Applied rewrites 57.9%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.0003968253968253968}, -2\right)\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*l* N/A

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

       3. lift-*.f64 N/A

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

       4. cube-unmult N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. cube-unmult N/A

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

       9. lift-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 57.9

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

   13. Applied rewrites 57.9%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 9.7

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

   5. Applied rewrites 9.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 23.8%

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

4. Final simplification 50.2%

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

Alternative 14: 47.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.4%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in re around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 57.5

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

   8. Applied rewrites 57.5%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 9.7

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

   5. Applied rewrites 9.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 23.8%

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

4. Final simplification 49.9%

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

Alternative 15: 46.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -2 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(-0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im\_m \cdot -0.16666666666666666\right), im\_m\right) \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -2e-6)
    (* im_m (* re (* -0.008333333333333333 (* (* im_m im_m) (* im_m im_m)))))
    (* (fma re (* re (* im_m -0.16666666666666666)) im_m) (- re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -2e-6) {
		tmp = im_m * (re * (-0.008333333333333333 * ((im_m * im_m) * (im_m * im_m))));
	} else {
		tmp = fma(re, (re * (im_m * -0.16666666666666666)), im_m) * -re;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -2e-6)
		tmp = Float64(im_m * Float64(re * Float64(-0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))));
	else
		tmp = Float64(fma(re, Float64(re * Float64(im_m * -0.16666666666666666)), im_m) * Float64(-re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-6], N[(im$95$m * N[(re * N[(-0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.99999999999999991e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 58.7

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

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

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

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

       5. lower-*.f64 N/A

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

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

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

       7. metadata-eval N/A

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

       8. lower-*.f64 N/A

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

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 58.7

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

   11. Applied rewrites 58.7%

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

   if -1.99999999999999991e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 49.6%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -2 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right) \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -2 \cdot 10^{-6}:\\ \;\;\;\;im\_m \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (sin re))) -2e-6)
    (* im_m (* -0.16666666666666666 (* re (* im_m im_m))))
    (* im_m (- re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -2e-6) {
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)));
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((exp(-im_m) - exp(im_m)) * (0.5d0 * sin(re))) <= (-2d-6)) then
        tmp = im_m * ((-0.16666666666666666d0) * (re * (im_m * im_m)))
    else
        tmp = im_m * -re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * Math.sin(re))) <= -2e-6) {
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)));
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if ((math.exp(-im_m) - math.exp(im_m)) * (0.5 * math.sin(re))) <= -2e-6:
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)))
	else:
		tmp = im_m * -re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * sin(re))) <= -2e-6)
		tmp = Float64(im_m * Float64(-0.16666666666666666 * Float64(re * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * Float64(-re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * sin(re))) <= -2e-6)
		tmp = im_m * (-0.16666666666666666 * (re * (im_m * im_m)));
	else
		tmp = im_m * -re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-6], N[(im$95$m * N[(-0.16666666666666666 * N[(re * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.99999999999999991e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. metadata-eval N/A

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

       5. sub-neg N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 51.0

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

   14. Applied rewrites 51.0%

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

   if -1.99999999999999991e-6 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 49.6%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 45.4

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

   8. Applied rewrites 45.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right) \leq -2 \cdot 10^{-6}:\\ \;\;\;\;im \cdot \left(-0.16666666666666666 \cdot \left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) 0.004)
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
        -0.3333333333333333)
       -2.0))
     (*
      re
      (fma
       (* re re)
       (fma
        re
        (* re (fma (* re re) -9.92063492063492e-5 0.004166666666666667))
        -0.08333333333333333)
       0.5)))
    (* im_m (* re (* im_m (* im_m -0.16666666666666666)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.004) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), fma(re, (re * fma((re * re), -9.92063492063492e-5, 0.004166666666666667)), -0.08333333333333333), 0.5));
	} else {
		tmp = im_m * (re * (im_m * (im_m * -0.16666666666666666)));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.004)
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), fma(re, Float64(re * fma(Float64(re * re), -9.92063492063492e-5, 0.004166666666666667)), -0.08333333333333333), 0.5)));
	else
		tmp = Float64(im_m * Float64(re * Float64(im_m * Float64(im_m * -0.16666666666666666))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 0.004], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5 + 0.004166666666666667), $MachinePrecision]), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 0.0040000000000000001

   1. Initial program 63.9%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 70.8

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   if 0.0040000000000000001 < (sin.f64 re)

   1. Initial program 55.0%

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 27.0

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

   8. Applied rewrites 27.0%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-in N/A

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

       4. metadata-eval N/A

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

       5. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

       12. lower-*.f64 26.9

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

   11. Applied rewrites 26.9%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]

   12. Taylor expanded in im around inf

       \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \frac{-1}{6}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot \frac{-1}{6}\right)\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{6} \cdot im\right)}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot im\right)\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}\right)\right) \]

       7. lower-*.f64 27.8

          \[\leadsto im \cdot \left(re \cdot \left(im \cdot \color{blue}{\left(im \cdot -0.16666666666666666\right)}\right)\right) \]

   14. Applied rewrites 27.8%

       \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.004:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;\left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im\_m \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) 0.0001)
    (*
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma
        (* im_m im_m)
        (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
        -0.3333333333333333)
       -2.0))
     (*
      re
      (fma (* re re) (* re (* re (* (* re re) -9.92063492063492e-5))) 0.5)))
    (*
     re
     (fma
      re
      (* re (* im_m (fma (* re re) -0.008333333333333333 0.16666666666666666)))
      (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.0001) {
		tmp = (im_m * fma((im_m * im_m), fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * (re * fma((re * re), (re * (re * ((re * re) * -9.92063492063492e-5))), 0.5));
	} else {
		tmp = re * fma(re, (re * (im_m * fma((re * re), -0.008333333333333333, 0.16666666666666666))), -im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.0001)
		tmp = Float64(Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333), -2.0)) * Float64(re * fma(Float64(re * re), Float64(re * Float64(re * Float64(Float64(re * re) * -9.92063492063492e-5))), 0.5)));
	else
		tmp = Float64(re * fma(re, Float64(re * Float64(im_m * fma(Float64(re * re), -0.008333333333333333, 0.16666666666666666))), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 0.0001], N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000005e-4

   1. Initial program 64.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. lower-*.f64 95.5

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Applied rewrites 95.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right)\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}\right) + \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)}\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, {re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) - \frac{1}{12}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)\right) + \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right), \frac{-1}{12}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{1}{240} + \frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2} + \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \left(\color{blue}{{re}^{2} \cdot \frac{-1}{10080}} + \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{10080}, \frac{1}{240}\right)}, \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{10080}, \frac{1}{240}\right), \frac{-1}{12}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      16. lower-*.f64 70.6

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \mathsf{fma}\left(re \cdot re, -9.92063492063492 \cdot 10^{-5}, 0.004166666666666667\right), -0.08333333333333333\right), 0.5\right)\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\frac{-1}{10080} \cdot {re}^{4}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot {re}^{\color{blue}{\left(2 \cdot 2\right)}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{10080} \cdot \color{blue}{\left({re}^{2} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot {re}^{2}}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(\left(\frac{-1}{10080} \cdot {re}^{2}\right) \cdot re\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{10080} \cdot {re}^{2}\right)\right)}, \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{10080}\right)}\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{-1}{10080}\right)\right), \frac{1}{2}\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

       14. lower-*.f64 70.3

           \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   11. Applied rewrites 70.3%

       \[\leadsto \left(re \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)}, 0.5\right)\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   if 1.00000000000000005e-4 < (sin.f64 re)

   1. Initial program 54.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 52.0

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right)\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right), \mathsf{neg}\left(im\right)\right)} \]

   8. Applied rewrites 30.2%

      \[\leadsto \color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \left(re \cdot \mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im\_m \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) 0.0001)
    (*
     re
     (*
      (fma re (* re -0.08333333333333333) 0.5)
      (*
       im_m
       (fma
        im_m
        (*
         im_m
         (fma
          (* im_m im_m)
          (fma (* im_m im_m) -0.0003968253968253968 -0.016666666666666666)
          -0.3333333333333333))
        -2.0))))
    (*
     re
     (fma
      re
      (* re (* im_m (fma (* re re) -0.008333333333333333 0.16666666666666666)))
      (- im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= 0.0001) {
		tmp = re * (fma(re, (re * -0.08333333333333333), 0.5) * (im_m * fma(im_m, (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333)), -2.0)));
	} else {
		tmp = re * fma(re, (re * (im_m * fma((re * re), -0.008333333333333333, 0.16666666666666666))), -im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= 0.0001)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.08333333333333333), 0.5) * Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.0003968253968253968, -0.016666666666666666), -0.3333333333333333)), -2.0))));
	else
		tmp = Float64(re * fma(re, Float64(re * Float64(im_m * fma(Float64(re * re), -0.008333333333333333, 0.16666666666666666))), Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], 0.0001], N[(re * N[(N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968 + -0.016666666666666666), $MachinePrecision] + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(re * N[(re * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-im$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000005e-4

   1. Initial program 64.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. lower-*.f64 95.5

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Applied rewrites 95.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left(im \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right) + \frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{12} \cdot im\right) \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} + \frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot im\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot {re}^{2}\right)} + \frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\left(\frac{-1}{12} \cdot im\right) \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right) \cdot {re}^{2}} + \frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{-1}{12} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right)} \cdot {re}^{2} + \frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)\right) \]

   8. Applied rewrites 70.6%

      \[\leadsto \color{blue}{re \cdot \left(\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]

   if 1.00000000000000005e-4 < (sin.f64 re)

   1. Initial program 54.2%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 52.0

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right)\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{120} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{6} \cdot im\right), \mathsf{neg}\left(im\right)\right)} \]

   8. Applied rewrites 30.2%

      \[\leadsto \color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 0.0001:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot \mathsf{fma}\left(re \cdot re, -0.008333333333333333, 0.16666666666666666\right)\right), -im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (*
     re
     (*
      (fma re (* re -0.08333333333333333) 0.5)
      (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))
    (*
     re
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * (fma(re, (re * -0.08333333333333333), 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0)));
	} else {
		tmp = re * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(fma(re, Float64(re * -0.08333333333333333), 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0))));
	else
		tmp = Float64(re * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(re * N[(N[(re * N[(re * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{3} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{3}\right)} + \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) + \color{blue}{-2}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{3}, -2\right)}\right) \]

      8. lower-*.f64 87.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot -0.3333333333333333}, -2\right)\right) \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.3333333333333333, -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left(im \cdot \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right) + \frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \frac{-1}{12} \cdot \left(im \cdot \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \frac{-1}{12} \cdot \left(im \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot {re}^{2}\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \frac{-1}{12} \cdot \color{blue}{\left(\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \cdot {re}^{2}\right)}\right) \]

      4. associate-*l* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \color{blue}{\left(\frac{-1}{12} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right) \cdot {re}^{2}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \left(\frac{-1}{12} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right) \cdot {re}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \color{blue}{{re}^{2} \cdot \left(\frac{-1}{12} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \color{blue}{\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{2} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) + \color{blue}{\left(\frac{-1}{12} \cdot {re}^{2}\right)} \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)\right) \]

   8. Applied rewrites 21.5%

      \[\leadsto \color{blue}{re \cdot \left(\left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right) \cdot \mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right)\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(im \cdot re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot im\right)} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      16. lower-*.f64 71.1

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   8. Applied rewrites 71.1%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(\mathsf{fma}\left(re, re \cdot -0.08333333333333333, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 56.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im\_m \cdot -0.16666666666666666\right), im\_m\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (fma re (* re (* im_m -0.16666666666666666)) im_m) (- re))
    (*
     re
     (*
      im_m
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = fma(re, (re * (im_m * -0.16666666666666666)), im_m) * -re;
	} else {
		tmp = re * (im_m * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * Float64(im_m * -0.16666666666666666)), im_m) * Float64(-re));
	else
		tmp = Float64(re * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision], N[(re * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right) + im\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot im\right) \cdot {re}^{2}} + im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{-1}{6} \cdot im\right)} + im\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot im\right) + im\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{6} \cdot im\right)\right)} + im\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{6} \cdot im\right), im\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot im\right)}, im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}, im\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto -re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot -0.16666666666666666\right)}, im\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto -\color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(im \cdot re\right) \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot im\right)} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto re \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      12. *-commutative N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      15. unpow2 N/A

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      16. lower-*.f64 71.1

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   8. Applied rewrites 71.1%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 56.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im\_m \cdot -0.16666666666666666\right), im\_m\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (fma re (* re (* im_m -0.16666666666666666)) im_m) (- re))
    (*
     im_m
     (*
      re
      (fma
       (* im_m im_m)
       (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
       -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = fma(re, (re * (im_m * -0.16666666666666666)), im_m) * -re;
	} else {
		tmp = im_m * (re * fma((im_m * im_m), fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * Float64(im_m * -0.16666666666666666)), im_m) * Float64(-re));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right) + im\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot im\right) \cdot {re}^{2}} + im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{-1}{6} \cdot im\right)} + im\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot im\right) + im\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{6} \cdot im\right)\right)} + im\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{6} \cdot im\right), im\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot im\right)}, im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}, im\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto -re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot -0.16666666666666666\right)}, im\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto -\color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      12. lower-*.f64 69.6

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   8. Applied rewrites 69.6%

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 55.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im\_m \cdot -0.16666666666666666\right), im\_m\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333 \cdot \left(im\_m \cdot im\_m\right), -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (fma re (* re (* im_m -0.16666666666666666)) im_m) (- re))
    (*
     im_m
     (*
      re
      (fma (* im_m im_m) (* -0.008333333333333333 (* im_m im_m)) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = fma(re, (re * (im_m * -0.16666666666666666)), im_m) * -re;
	} else {
		tmp = im_m * (re * fma((im_m * im_m), (-0.008333333333333333 * (im_m * im_m)), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * Float64(im_m * -0.16666666666666666)), im_m) * Float64(-re));
	else
		tmp = Float64(im_m * Float64(re * fma(Float64(im_m * im_m), Float64(-0.008333333333333333 * Float64(im_m * im_m)), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision], N[(im$95$m * N[(re * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right) + im\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot im\right) \cdot {re}^{2}} + im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{-1}{6} \cdot im\right)} + im\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot im\right) + im\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{6} \cdot im\right)\right)} + im\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{6} \cdot im\right), im\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot im\right)}, im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}, im\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto -re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot -0.16666666666666666\right)}, im\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto -\color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto im \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto im \cdot \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{-1}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}, -1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, -1\right)\right) \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), -1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{120} + \color{blue}{\frac{-1}{6}}, -1\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{120}, \frac{-1}{6}\right)}, -1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{120}, \frac{-1}{6}\right), -1\right)\right) \]

      12. lower-*.f64 69.6

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right) \]

   8. Applied rewrites 69.6%

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), -1\right)\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{120} \cdot {im}^{2}}, -1\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}}, -1\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{120}}, -1\right)\right) \]

       3. unpow2 N/A

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{120}, -1\right)\right) \]

       4. lower-*.f64 69.5

          \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot -0.008333333333333333, -1\right)\right) \]

   11. Applied rewrites 69.5%

       \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right) \cdot -0.008333333333333333}, -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, -0.008333333333333333 \cdot \left(im \cdot im\right), -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 53.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im\_m \cdot -0.16666666666666666\right), im\_m\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* (fma re (* re (* im_m -0.16666666666666666)) im_m) (- re))
    (* re (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = fma(re, (re * (im_m * -0.16666666666666666)), im_m) * -re;
	} else {
		tmp = re * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(fma(re, Float64(re * Float64(im_m * -0.16666666666666666)), im_m) * Float64(-re));
	else
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + im$95$m), $MachinePrecision] * (-re)), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right) + im\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot im\right) \cdot {re}^{2}} + im\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{{re}^{2} \cdot \left(\frac{-1}{6} \cdot im\right)} + im\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(\frac{-1}{6} \cdot im\right) + im\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \left(\color{blue}{re \cdot \left(re \cdot \left(\frac{-1}{6} \cdot im\right)\right)} + im\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \color{blue}{\mathsf{fma}\left(re, re \cdot \left(\frac{-1}{6} \cdot im\right), im\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, \color{blue}{re \cdot \left(\frac{-1}{6} \cdot im\right)}, im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)}, im\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto -re \cdot \mathsf{fma}\left(re, re \cdot \color{blue}{\left(im \cdot -0.16666666666666666\right)}, im\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto -\color{blue}{re \cdot \mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. lower-*.f64 93.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Applied rewrites 93.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 73.2

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + -1 \cdot re\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re} + -1 \cdot re\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

       12. lower-*.f64 65.1

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

   11. Applied rewrites 65.1%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)} + -1\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right) \cdot re\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right)\right) \cdot re} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right) \cdot im\right)} \cdot re \]

       6. lift-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)} \cdot im\right) \cdot re \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \frac{-1}{6}} + -1\right) \cdot im\right) \cdot re \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + -1\right) \cdot im\right) \cdot re \]

       9. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + -1\right) \cdot im\right) \cdot re \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)} + -1\right) \cdot im\right) \cdot re \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)} \cdot im\right) \cdot re \]

       12. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right) \cdot im\right) \cdot re} \]

   13. Applied rewrites 68.6%

       \[\leadsto \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot \left(im \cdot -0.16666666666666666\right), im\right) \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 53.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* re (* im_m (fma 0.16666666666666666 (* re re) -1.0)))
    (* re (* im_m (fma im_m (* im_m -0.16666666666666666) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * (im_m * fma(0.16666666666666666, (re * re), -1.0));
	} else {
		tmp = re * (im_m * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(im_m * fma(0.16666666666666666, Float64(re * re), -1.0)));
	else
		tmp = Float64(re * Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(re * N[(im$95$m * N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {re}^{2}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{re \cdot re}, -1\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{re \cdot re}, -1\right)\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. lower-*.f64 93.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Applied rewrites 93.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 73.2

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + -1 \cdot re\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re} + -1 \cdot re\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

       12. lower-*.f64 65.1

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

   11. Applied rewrites 65.1%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(im \cdot im\right)} + -1\right)\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right) \cdot re\right)} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(im \cdot \mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right)\right) \cdot re} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{6}, im \cdot im, -1\right) \cdot im\right)} \cdot re \]

       6. lift-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(im \cdot im\right) + -1\right)} \cdot im\right) \cdot re \]

       7. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(im \cdot im\right) \cdot \frac{-1}{6}} + -1\right) \cdot im\right) \cdot re \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{6} + -1\right) \cdot im\right) \cdot re \]

       9. associate-*r* N/A

          \[\leadsto \left(\left(\color{blue}{im \cdot \left(im \cdot \frac{-1}{6}\right)} + -1\right) \cdot im\right) \cdot re \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)} + -1\right) \cdot im\right) \cdot re \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)} \cdot im\right) \cdot re \]

       12. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right) \cdot im\right) \cdot re} \]

   13. Applied rewrites 68.6%

       \[\leadsto \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 50.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.02:\\ \;\;\;\;re \cdot \left(im\_m \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.02)
    (* re (* im_m (fma 0.16666666666666666 (* re re) -1.0)))
    (* im_m (* re (fma -0.16666666666666666 (* im_m im_m) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.02) {
		tmp = re * (im_m * fma(0.16666666666666666, (re * re), -1.0));
	} else {
		tmp = im_m * (re * fma(-0.16666666666666666, (im_m * im_m), -1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.02)
		tmp = Float64(re * Float64(im_m * fma(0.16666666666666666, Float64(re * re), -1.0)));
	else
		tmp = Float64(im_m * Float64(re * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.02], N[(re * N[(im$95$m * N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0200000000000000004

   1. Initial program 50.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

      4. lower-sin.f64 56.3

         \[\leadsto -im \cdot \color{blue}{\sin re} \]

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{-im \cdot \sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]

      2. sub-neg N/A

         \[\leadsto re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto re \cdot \left(\frac{1}{6} \cdot \color{blue}{\left({re}^{2} \cdot im\right)} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto re \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im} + \left(\mathsf{neg}\left(im\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto re \cdot \left(\left(\frac{1}{6} \cdot {re}^{2}\right) \cdot im + \color{blue}{-1 \cdot im}\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{6} \cdot {re}^{2} + -1\right)\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto re \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {re}^{2}, -1\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{re \cdot re}, -1\right)\right) \]

      10. lower-*.f64 18.6

          \[\leadsto re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, \color{blue}{re \cdot re}, -1\right)\right) \]

   8. Applied rewrites 18.6%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right)\right)} \]

   if -0.0200000000000000004 < (sin.f64 re)

   1. Initial program 65.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

      17. lower-*.f64 93.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   5. Applied rewrites 93.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 73.2

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   8. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + -1 \cdot re\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re} + -1 \cdot re\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right)} \]

       4. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       5. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

       8. sub-neg N/A

          \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

       9. metadata-eval N/A

          \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

       11. unpow2 N/A

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

       12. lower-*.f64 65.1

           \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

   11. Applied rewrites 65.1%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 51.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* re (fma -0.16666666666666666 (* im_m im_m) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * fma(-0.16666666666666666, (im_m * im_m), -1.0)));
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(re * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0))))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(re * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{-2}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}, -2\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, -2\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\frac{-1}{3}}, -2\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right)}, -2\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}, \frac{-1}{3}\right), -2\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{2520} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}, \frac{-1}{3}\right), -2\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{2520}} + \left(\mathsf{neg}\left(\frac{1}{60}\right)\right), \frac{-1}{3}\right), -2\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \frac{-1}{2520} + \color{blue}{\frac{-1}{60}}, \frac{-1}{3}\right), -2\right)\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{2520}, \frac{-1}{60}\right)}, \frac{-1}{3}\right), -2\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]

   17. lower-*.f64 93.6

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

5. Applied rewrites 93.6%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{-1}{2520}, \frac{-1}{60}\right), \frac{-1}{3}\right), -2\right)\right) \]
7. Step-by-step derivation

   1. lower-*.f64 61.0

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

8. Applied rewrites 61.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right), -0.3333333333333333\right), -2\right)\right) \]

9. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + -1 \cdot re\right)} \]

    2. associate-*r* N/A

       \[\leadsto im \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re} + -1 \cdot re\right) \]

    3. distribute-rgt-in N/A

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right)} \]

    4. metadata-eval N/A

       \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

    5. sub-neg N/A

       \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto im \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)\right)} \]

    8. sub-neg N/A

       \[\leadsto im \cdot \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \]

    9. metadata-eval N/A

       \[\leadsto im \cdot \left(re \cdot \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{-1}\right)\right) \]

    10. lower-fma.f64 N/A

        \[\leadsto im \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {im}^{2}, -1\right)}\right) \]

    11. unpow2 N/A

        \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{im \cdot im}, -1\right)\right) \]

    12. lower-*.f64 55.0

        \[\leadsto im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{im \cdot im}, -1\right)\right) \]

11. Applied rewrites 55.0%

    \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\right)} \]
12. Add Preprocessing

Alternative 28: 33.4% accurate, 39.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m (- re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -re);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * -re)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -re);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * -re)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(-re)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -re);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * (-re)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot \sin re\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{im \cdot \sin re}\right) \]

   4. lower-sin.f64 56.5

      \[\leadsto -im \cdot \color{blue}{\sin re} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{-im \cdot \sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(im \cdot re\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{re \cdot im}\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{re \cdot \left(\mathsf{neg}\left(im\right)\right)} \]

   5. lower-neg.f64 37.5

      \[\leadsto re \cdot \color{blue}{\left(-im\right)} \]

8. Applied rewrites 37.5%

   \[\leadsto \color{blue}{re \cdot \left(-im\right)} \]

9. Final simplification 37.5%

   \[\leadsto im \cdot \left(-re\right) \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024219 
(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))))