math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.8%

Time: 10.1s

Alternatives: 23

Speedup: 2.5×

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

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - e^{im\_m} \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5\right) \cdot e^{im\_m}, \cos re, \left(\cos re \cdot 0.5\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot \left(\cos re \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.2)
      (fma (* (- 0.5) (exp im_m)) (cos re) (* (* (cos re) 0.5) t_0))
      (* (fma (* -0.16666666666666666 im_m) im_m -1.0) (* (cos re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.2) {
		tmp = fma((-0.5 * exp(im_m)), cos(re), ((cos(re) * 0.5) * t_0));
	} else {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * (cos(re) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.2)
		tmp = fma(Float64(Float64(-0.5) * exp(im_m)), cos(re), Float64(Float64(cos(re) * 0.5) * t_0));
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * Float64(cos(re) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.2], N[(N[((-0.5) * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision] + N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

      3. sub-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{im}\right)\right) + e^{0 - im}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right) \cdot \cos re} + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-e^{im}\right)} \cdot \frac{1}{2}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot e^{0 - im}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{e^{0 - im}}\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{\color{blue}{0 - im}}\right) \]

      14. exp-diff N/A

          \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\frac{e^{0}}{e^{im}}}\right) \]

      15. exp-0 N/A

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

      16. lift-exp.f64 N/A

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

      17. un-div-inv N/A

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

      18. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-e^{im}\right) \cdot 0.5, \cos re, \frac{\cos re \cdot 0.5}{e^{im}}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. exp-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-*.f64 100.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 36.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

   8. Applied rewrites 89.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5\right) \cdot e^{im}, \cos re, \left(\cos re \cdot 0.5\right) \cdot e^{-im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ t_1 := e^{-im\_m} - e^{im\_m}\\ t_2 := \left(\cos re \cdot 0.5\right) \cdot t\_1\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_1 \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot \left(\cos re \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im\_m}^{4}, \mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot t\_0, \mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot t\_0\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* re re) -0.25 0.5))
        (t_1 (- (exp (- im_m)) (exp im_m)))
        (t_2 (* (* (cos re) 0.5) t_1)))
   (*
    im_s
    (if (<= t_2 -0.1)
      (* t_1 0.5)
      (if (<= t_2 2e-5)
        (* (fma (* -0.16666666666666666 im_m) im_m -1.0) (* (cos re) im_m))
        (*
         (fma
          (pow im_m 4.0)
          (*
           (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
           t_0)
          (* (fma -0.3333333333333333 (* im_m im_m) -2.0) t_0))
         im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((re * re), -0.25, 0.5);
	double t_1 = exp(-im_m) - exp(im_m);
	double t_2 = (cos(re) * 0.5) * t_1;
	double tmp;
	if (t_2 <= -0.1) {
		tmp = t_1 * 0.5;
	} else if (t_2 <= 2e-5) {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * (cos(re) * im_m);
	} else {
		tmp = fma(pow(im_m, 4.0), (fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * t_0), (fma(-0.3333333333333333, (im_m * im_m), -2.0) * t_0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(re * re), -0.25, 0.5)
	t_1 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_2 = Float64(Float64(cos(re) * 0.5) * t_1)
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(t_1 * 0.5);
	elseif (t_2 <= 2e-5)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * Float64(cos(re) * im_m));
	else
		tmp = Float64(fma((im_m ^ 4.0), Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * t_0), Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * t_0)) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, -0.1], N[(t$95$1 * 0.5), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im$95$m, 4.0], $MachinePrecision] * N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 7.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 76.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(-2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{60} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + \frac{-1}{2520} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 68.3%

      \[\leadsto \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot \left(\cos re \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right), \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ t_2 := e^{-im\_m}\\ t_3 := t\_0 \cdot \left(t\_2 - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\log t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im\_m, im\_m, -0.016666666666666666\right) \cdot im\_m, im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im\_m}^{4}, \mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot t\_1, \mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot t\_1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (cos re) 0.5))
        (t_1 (fma (* re re) -0.25 0.5))
        (t_2 (exp (- im_m)))
        (t_3 (* t_0 (- t_2 (exp im_m)))))
   (*
    im_s
    (if (<= t_3 (- INFINITY))
      (log t_2)
      (if (<= t_3 2e-5)
        (*
         (*
          (fma
           (fma
            (*
             (fma (* -0.0003968253968253968 im_m) im_m -0.016666666666666666)
             im_m)
            im_m
            -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m)
         t_0)
        (*
         (fma
          (pow im_m 4.0)
          (*
           (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
           t_1)
          (* (fma -0.3333333333333333 (* im_m im_m) -2.0) t_1))
         im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double t_1 = fma((re * re), -0.25, 0.5);
	double t_2 = exp(-im_m);
	double t_3 = t_0 * (t_2 - exp(im_m));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = log(t_2);
	} else if (t_3 <= 2e-5) {
		tmp = (fma(fma((fma((-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	} else {
		tmp = fma(pow(im_m, 4.0), (fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * t_1), (fma(-0.3333333333333333, (im_m * im_m), -2.0) * t_1)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = fma(Float64(re * re), -0.25, 0.5)
	t_2 = exp(Float64(-im_m))
	t_3 = Float64(t_0 * Float64(t_2 - exp(im_m)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = log(t_2);
	elseif (t_3 <= 2e-5)
		tmp = Float64(Float64(fma(fma(Float64(fma(Float64(-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	else
		tmp = Float64(fma((im_m ^ 4.0), Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * t_1), Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * t_1)) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-im$95$m)], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(t$95$2 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$3, (-Infinity)], N[Log[t$95$2], $MachinePrecision], If[LessEqual[t$95$3, 2e-5], N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Power[im$95$m, 4.0], $MachinePrecision] * N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.9%

      \[\leadsto -im \]
   9. Step-by-step derivation

   10. Applied rewrites 72.1%

       \[\leadsto \log \left(e^{-im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 9.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 76.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(-2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{60} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + \frac{-1}{2520} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 68.3%

      \[\leadsto \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\log \left(e^{-im}\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im, im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right), \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ t_1 := \cos re \cdot 0.5\\ t_2 := t\_1 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_3 := \mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_3, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im\_m, im\_m, -0.016666666666666666\right) \cdot im\_m, im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im\_m}^{4}, t\_3 \cdot t\_0, \mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot t\_0\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* re re) -0.25 0.5))
        (t_1 (* (cos re) 0.5))
        (t_2 (* t_1 (- (exp (- im_m)) (exp im_m))))
        (t_3 (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)))
   (*
    im_s
    (if (<= t_2 (- INFINITY))
      (*
       (*
        (fma (fma t_3 (* im_m im_m) -0.3333333333333333) (* im_m im_m) -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (if (<= t_2 2e-5)
        (*
         (*
          (fma
           (fma
            (*
             (fma (* -0.0003968253968253968 im_m) im_m -0.016666666666666666)
             im_m)
            im_m
            -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m)
         t_1)
        (*
         (fma
          (pow im_m 4.0)
          (* t_3 t_0)
          (* (fma -0.3333333333333333 (* im_m im_m) -2.0) t_0))
         im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((re * re), -0.25, 0.5);
	double t_1 = cos(re) * 0.5;
	double t_2 = t_1 * (exp(-im_m) - exp(im_m));
	double t_3 = fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (fma(fma(t_3, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else if (t_2 <= 2e-5) {
		tmp = (fma(fma((fma((-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_1;
	} else {
		tmp = fma(pow(im_m, 4.0), (t_3 * t_0), (fma(-0.3333333333333333, (im_m * im_m), -2.0) * t_0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(re * re), -0.25, 0.5)
	t_1 = Float64(cos(re) * 0.5)
	t_2 = Float64(t_1 * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_3 = fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(t_3, Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	elseif (t_2 <= 2e-5)
		tmp = Float64(Float64(fma(fma(Float64(fma(Float64(-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_1);
	else
		tmp = Float64(fma((im_m ^ 4.0), Float64(t_3 * t_0), Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * t_0)) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[(t$95$3 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Power[im$95$m, 4.0], $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 68.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 9.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 76.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left(-2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + {im}^{2} \cdot \left(\frac{-1}{60} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) + \frac{-1}{2520} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 68.3%

      \[\leadsto \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im, im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right), \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot \left(\cos re \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (cos re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -0.1)
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (if (<= t_0 2e-5)
        (* (fma (* -0.16666666666666666 im_m) im_m -1.0) (* (cos re) im_m))
        (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (cos(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -0.1) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else if (t_0 <= 2e-5) {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * (cos(re) * im_m);
	} else {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	elseif (t_0 <= 2e-5)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * Float64(cos(re) * im_m));
	else
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.1], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 66.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.3%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 7.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

   8. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 76.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.2%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 25.2%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot \left(\cos re \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (cos re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -4e-5)
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (if (<= t_0 2e-5)
        (* (- (cos re)) im_m)
        (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (cos(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -4e-5) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else if (t_0 <= 2e-5) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -4e-5)
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	elseif (t_0 <= 2e-5)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -4e-5], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -4.00000000000000033e-5

   1. Initial program 99.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 66.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -4.00000000000000033e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.00000000000000016e-5

   1. Initial program 6.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2.00000000000000016e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 76.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.2%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 25.2%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im\_m, im\_m, -0.016666666666666666\right) \cdot im\_m, im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (cos re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (if (<= t_0 0.0)
        (*
         0.5
         (*
          (fma
           (fma
            (*
             (fma (* -0.0003968253968253968 im_m) im_m -0.016666666666666666)
             im_m)
            im_m
            -0.3333333333333333)
           (* im_m im_m)
           -2.0)
          im_m))
        (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (cos(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else if (t_0 <= 0.0) {
		tmp = 0.5 * (fma(fma((fma((-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	} else {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(cos(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	elseif (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(fma(fma(Float64(fma(Float64(-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	else
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 68.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 8.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 53.1%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 98.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 74.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 58.0%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.5%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 24.5%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im, im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\cos re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(-im\_m\right) \cdot im\_m}{im\_m}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (cos re) 0.5) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (/ (* (- im_m) im_m) im_m)
      (if (<= t_0 0.0) (- im_m) (* (* (* re 0.5) re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (cos(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-im_m * im_m) / im_m;
	} else if (t_0 <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = ((re * 0.5) * re) * im_m;
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = (Math.cos(re) * 0.5) * (Math.exp(-im_m) - Math.exp(im_m));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-im_m * im_m) / im_m;
	} else if (t_0 <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = ((re * 0.5) * re) * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (math.cos(re) * 0.5) * (math.exp(-im_m) - math.exp(im_m))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-im_m * im_m) / im_m
	elif t_0 <= 0.0:
		tmp = -im_m
	else:
		tmp = ((re * 0.5) * re) * im_m
	return im_s * tmp

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = (cos(re) * 0.5) * (exp(-im_m) - exp(im_m));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-im_m * im_m) / im_m;
	elseif (t_0 <= 0.0)
		tmp = -im_m;
	else
		tmp = ((re * 0.5) * re) * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[((-im$95$m) * im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-im$95$m), N[(N[(N[(re * 0.5), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.9%

      \[\leadsto -im \]
   9. Step-by-step derivation

   10. Applied rewrites 39.6%

       \[\leadsto \frac{-im \cdot im}{im} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 8.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.4%

      \[\leadsto -im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 98.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 74.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 58.0%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 24.5%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 18.4%

       \[\leadsto \left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\frac{\left(-im\right) \cdot im}{im}\\ \mathbf{elif}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right) \cdot \left(\cos re \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.2)
      (* (* (cos re) 0.5) t_0)
      (* (fma (* -0.16666666666666666 im_m) im_m -1.0) (* (cos re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (cos(re) * 0.5) * t_0;
	} else {
		tmp = fma((-0.16666666666666666 * im_m), im_m, -1.0) * (cos(re) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	else
		tmp = Float64(fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0) * Float64(cos(re) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lower-*.f64 100.0

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. sub0-neg N/A

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

      6. lower-neg.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 36.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

   8. Applied rewrites 89.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \cdot \left(\cos re \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im\_m, im\_m, -0.016666666666666666\right) \cdot im\_m, im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (*
       (*
        (fma
         (fma
          (*
           (fma (* -0.0003968253968253968 im_m) im_m -0.016666666666666666)
           im_m)
          im_m
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else {
		tmp = (fma(fma((fma((-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	else
		tmp = Float64(Float64(fma(fma(Float64(fma(Float64(-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 68.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

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

   1. Initial program 36.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 92.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im, im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (*
       (*
        (fma
         (fma
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5))
      (*
       (*
        (fma
         (fma -0.016666666666666666 (* im_m im_m) -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * fma((0.020833333333333332 * (re * re)), (re * re), 0.5);
	} else {
		tmp = (fma(fma(-0.016666666666666666, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5));
	else
		tmp = Float64(Float64(fma(fma(-0.016666666666666666, Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 68.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2}, \color{blue}{re} \cdot re, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), \color{blue}{re} \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

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

   1. Initial program 36.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -5 \cdot 10^{+99}:\\ \;\;\;\;t\_0 \cdot \left(\left(1 - im\_m\right) - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im\_m, im\_m, -0.016666666666666666\right) \cdot im\_m, im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)))
   (*
    im_s
    (if (<= (- (exp (- im_m)) (exp im_m)) -5e+99)
      (* t_0 (- (- 1.0 im_m) (exp im_m)))
      (*
       (*
        (fma
         (fma
          (*
           (fma (* -0.0003968253968253968 im_m) im_m -0.016666666666666666)
           im_m)
          im_m
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -5e+99) {
		tmp = t_0 * ((1.0 - im_m) - exp(im_m));
	} else {
		tmp = (fma(fma((fma((-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -5e+99)
		tmp = Float64(t_0 * Float64(Float64(1.0 - im_m) - exp(im_m)));
	else
		tmp = Float64(Float64(fma(fma(Float64(fma(Float64(-0.0003968253968253968 * im_m), im_m, -0.016666666666666666) * im_m), im_m, -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -5e+99], N[(t$95$0 * N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * im$95$m), $MachinePrecision] * im$95$m + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -5.00000000000000008e99

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lower-*.f64 100.0

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \left(e^{\color{blue}{0 - im}} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

      5. sub0-neg N/A

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

      6. lower-neg.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -5.00000000000000008e99 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 37.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 94.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(\left(1 - im\right) - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot im, im, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\cos re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \mathbf{elif}\;\cos re \leq 0.926:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right) \cdot im\_m\right) \cdot re, re, -im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot 0.5\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) 0.004)
    (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m)
    (if (<= (cos re) 0.926)
      (fma
       (* (* (fma -0.041666666666666664 (* re re) 0.5) im_m) re)
       re
       (- im_m))
      (* (* (fma (* im_m im_m) -0.3333333333333333 -2.0) 0.5) 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 (cos(re) <= 0.004) {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	} else if (cos(re) <= 0.926) {
		tmp = fma(((fma(-0.041666666666666664, (re * re), 0.5) * im_m) * re), re, -im_m);
	} else {
		tmp = (fma((im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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 (cos(re) <= 0.004)
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	elseif (cos(re) <= 0.926)
		tmp = fma(Float64(Float64(fma(-0.041666666666666664, Float64(re * re), 0.5) * im_m) * re), re, Float64(-im_m));
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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[Cos[re], $MachinePrecision], 0.004], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.926], N[(N[(N[(N[(-0.041666666666666664 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision] * re + (-im$95$m)), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < 0.0040000000000000001

   1. Initial program 49.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 46.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.3%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.2%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.2%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if 0.0040000000000000001 < (cos.f64 re) < 0.926000000000000045

   1. Initial program 52.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im + \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{2} \cdot im\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto \mathsf{fma}\left(\left(im \cdot \mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right)\right) \cdot re, \color{blue}{re}, -im\right) \]

   if 0.926000000000000045 < (cos.f64 re)

   1. Initial program 55.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 50.2

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 80.4%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 85.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;\cos re \leq 0.926:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right) \cdot im\right) \cdot re, re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \mathbf{elif}\;\cos re \leq 0.926:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right) \cdot 0.5\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) 0.004)
    (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m)
    (if (<= (cos re) 0.926)
      (* (fma (fma -0.041666666666666664 (* re re) 0.5) (* re re) -1.0) im_m)
      (* (* (fma (* im_m im_m) -0.3333333333333333 -2.0) 0.5) 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 (cos(re) <= 0.004) {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	} else if (cos(re) <= 0.926) {
		tmp = fma(fma(-0.041666666666666664, (re * re), 0.5), (re * re), -1.0) * im_m;
	} else {
		tmp = (fma((im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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 (cos(re) <= 0.004)
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	elseif (cos(re) <= 0.926)
		tmp = Float64(fma(fma(-0.041666666666666664, Float64(re * re), 0.5), Float64(re * re), -1.0) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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[Cos[re], $MachinePrecision], 0.004], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.926], N[(N[(N[(-0.041666666666666664 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < 0.0040000000000000001

   1. Initial program 49.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 46.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.3%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.2%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.2%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if 0.0040000000000000001 < (cos.f64 re) < 0.926000000000000045

   1. Initial program 52.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {re}^{2}\right) - 1\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, -1\right) \cdot im \]

   if 0.926000000000000045 < (cos.f64 re)

   1. Initial program 55.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 50.2

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 80.4%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 85.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \mathbf{elif}\;\cos re \leq 0.926:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, -1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot re\right)\right) \cdot re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) 0.004)
    (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m)
    (*
     (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
     (*
      (fma
       (fma
        (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
        (* im_m im_m)
        -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= 0.004) {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	} else {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * (fma(fma(fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m);
	}
	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 (cos(re) <= 0.004)
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	else
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * Float64(fma(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], 0.004], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.0040000000000000001

   1. Initial program 49.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 46.8

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.3%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.2%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.2%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if 0.0040000000000000001 < (cos.f64 re)

   1. Initial program 54.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{48} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{48} \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right), im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]

      10. lower-*.f64 79.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   8. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.004:\\ \;\;\;\;\left(\left(0.08333333333333333 \cdot \left(\left(im \cdot im\right) \cdot re\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 47.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.8%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.7%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0003968253968253968, -0.016666666666666666\right) \cdot im, im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 91.2%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 78.6%

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

4. Final simplification 69.9%

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

Alternative 17: 69.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 47.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.8%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.7%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 75.8%

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

4. Final simplification 67.9%

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

Alternative 18: 64.1% accurate, 2.4× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -0.01)
    (* (* (* 0.08333333333333333 (* (* im_m im_m) re)) re) im_m)
    (* (* (fma (* im_m im_m) -0.3333333333333333 -2.0) 0.5) 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 (cos(re) <= -0.01) {
		tmp = ((0.08333333333333333 * ((im_m * im_m) * re)) * re) * im_m;
	} else {
		tmp = (fma((im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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 (cos(re) <= -0.01)
		tmp = Float64(Float64(Float64(0.08333333333333333 * Float64(Float64(im_m * im_m) * re)) * re) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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[Cos[re], $MachinePrecision], -0.01], N[(N[(N[(0.08333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 47.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.8%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 47.7%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot re\right) \cdot 0.08333333333333333\right) \cdot re\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 37.6

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 71.1%

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

4. Final simplification 64.5%

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

Alternative 19: 62.3% accurate, 2.5× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -0.01)
    (* (fma (* re re) 0.5 -1.0) im_m)
    (* (* (fma (* im_m im_m) -0.3333333333333333 -2.0) 0.5) 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 (cos(re) <= -0.01) {
		tmp = fma((re * re), 0.5, -1.0) * im_m;
	} else {
		tmp = (fma((im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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 (cos(re) <= -0.01)
		tmp = Float64(fma(Float64(re * re), 0.5, -1.0) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), -0.3333333333333333, -2.0) * 0.5) * 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[Cos[re], $MachinePrecision], -0.01], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.0%

      \[\leadsto \mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 37.6

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 71.1%

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

Alternative 20: 38.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= (cos re) -0.01) (* (fma (* re re) 0.5 -1.0) im_m) (- 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 (cos(re) <= -0.01) {
		tmp = fma((re * re), 0.5, -1.0) * im_m;
	} else {
		tmp = -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 (cos(re) <= -0.01)
		tmp = Float64(fma(Float64(re * re), 0.5, -1.0) * im_m);
	else
		tmp = 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[Cos[re], $MachinePrecision], -0.01], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision], (-im$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.0%

      \[\leadsto \mathsf{fma}\left(re \cdot re, 0.5, -1\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.9%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 38.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= (cos re) -0.01) (* (* (* re 0.5) re) im_m) (- 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 (cos(re) <= -0.01) {
		tmp = ((re * 0.5) * re) * im_m;
	} else {
		tmp = -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (cos(re) <= (-0.01d0)) then
        tmp = ((re * 0.5d0) * re) * im_m
    else
        tmp = -im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (Math.cos(re) <= -0.01) {
		tmp = ((re * 0.5) * re) * im_m;
	} else {
		tmp = -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if math.cos(re) <= -0.01:
		tmp = ((re * 0.5) * re) * im_m
	else:
		tmp = -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 (cos(re) <= -0.01)
		tmp = Float64(Float64(Float64(re * 0.5) * re) * im_m);
	else
		tmp = Float64(-im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (cos(re) <= -0.01)
		tmp = ((re * 0.5) * re) * im_m;
	else
		tmp = -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(N[(re * 0.5), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], (-im$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 50.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

      14. lower-exp.f64 47.3

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - \color{blue}{e^{im}}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 46.8%

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right)\right) \cdot \color{blue}{im} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 46.8%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot -0.25\right) \cdot re\right) \cdot re\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 39.0%

       \[\leadsto \left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im \]

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 54.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto -1 \cdot \color{blue}{im} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.9%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\left(re \cdot 0.5\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.1% accurate, 105.7× speedup

Math

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

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)))

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;
}

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
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;
}

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

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))
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;
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 * (-im$95$m)), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cos.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in re around 0

   \[\leadsto -1 \cdot \color{blue}{im} \]
7. Step-by-step derivation

8. Applied rewrites 28.5%

   \[\leadsto -im \]
9. Add Preprocessing

Alternative 23: 3.4% accurate, 317.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 0 \end{array} \]

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 0.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 53.1%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} - e^{im}\right)} \]

   3. sub-neg N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(\mathsf{neg}\left(e^{im}\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(e^{im}\right)\right) + e^{0 - im}\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \left(\frac{1}{2} \cdot \cos re\right) + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right) \cdot \cos re} + e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(-e^{im}\right)} \cdot \frac{1}{2}, \cos re, e^{0 - im} \cdot \left(\frac{1}{2} \cdot \cos re\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot e^{0 - im}}\right) \]

   12. lift-exp.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{e^{0 - im}}\right) \]

   13. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot e^{\color{blue}{0 - im}}\right) \]

   14. exp-diff N/A

       \[\leadsto \mathsf{fma}\left(\left(-e^{im}\right) \cdot \frac{1}{2}, \cos re, \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\frac{e^{0}}{e^{im}}}\right) \]

   15. exp-0 N/A

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

   16. lift-exp.f64 N/A

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

   17. un-div-inv N/A

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

   18. lower-/.f64 53.1

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

4. Applied rewrites 53.1%

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

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \cos re + \frac{1}{2} \cdot \cos re} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \cos re \cdot \color{blue}{0} \]

   3. mul0-rgt 3.6

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 3.6%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - 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.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(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 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[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[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024288 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))