math.sin on complex, imaginary part

Percentage Accurate: 53.3% → 99.4%

Time: 13.2s

Alternatives: 17

Speedup: 2.6×

• 48.7% 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: 53.3% 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: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.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

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

   1. Initial program 35.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.3

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

   5. Simplified 88.3%

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

4. Final simplification 91.0%

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

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* 0.5 (- 1.0 (exp im_m)))
      (if (<= t_0 0.0)
        (* im_m (* (cos re) (fma im_m (* im_m -0.16666666666666666) -1.0)))
        (*
         (fma (* re re) -0.25 0.5)
         (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else if (t_0 <= 0.0) {
		tmp = im_m * (cos(re) * fma(im_m, (im_m * -0.16666666666666666), -1.0));
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(cos(re) * fma(im_m, Float64(im_m * -0.16666666666666666), -1.0)));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   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))) < 0.0

   1. Initial program 6.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \left(\cos re \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* 0.5 (- 1.0 (exp im_m)))
      (if (<= t_0 0.0)
        (* (cos re) (- im_m))
        (*
         (fma (* re re) -0.25 0.5)
         (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else if (t_0 <= 0.0) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 0.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   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))) < 0.0

   1. Initial program 6.2%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.6

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

   5. Simplified 99.6%

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

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right)\right), im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e-5)
      (*
       im_m
       (*
        (fma (* re re) (fma re (* re 0.041666666666666664) -0.5) 1.0)
        (fma
         (*
          (* im_m im_m)
          (*
           im_m
           (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)))
         im_m
         (fma im_m (* im_m -0.16666666666666666) -1.0))))
      (if (<= t_0 0.0)
        (* (cos re) (- im_m))
        (*
         (fma (* re re) -0.25 0.5)
         (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e-5) {
		tmp = im_m * (fma((re * re), fma(re, (re * 0.041666666666666664), -0.5), 1.0) * fma(((im_m * im_m) * (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333))), im_m, fma(im_m, (im_m * -0.16666666666666666), -1.0)));
	} else if (t_0 <= 0.0) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e-5)
		tmp = Float64(im_m * Float64(fma(Float64(re * re), fma(re, Float64(re * 0.041666666666666664), -0.5), 1.0) * fma(Float64(Float64(im_m * im_m) * Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333))), im_m, fma(im_m, Float64(im_m * -0.16666666666666666), -1.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e-5], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im$95$m + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $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))) < -2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. lower-*.f64 69.8

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.041666666666666664}, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   8. Simplified 69.8%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right)} \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)} + -1\right)\right)\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot im} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot im\right), im, \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)}\right) \]

   10. Applied egg-rr 69.8%

       \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right)\right) \cdot \left(im \cdot im\right), im, \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)}\right) \]

   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))) < 0.0

   1. Initial program 5.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.8

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

   5. Simplified 99.8%

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

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right)\right), im, \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ t_1 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(t\_1, im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right), -1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot t\_1, -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))))
        (t_1 (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (*
        (fma (* re re) (fma re (* re 0.041666666666666664) -0.5) 1.0)
        (fma t_1 (* im_m (* im_m (* im_m im_m))) -1.0)))
      (if (<= t_0 0.0)
        (*
         im_m
         (fma (* im_m im_m) (fma im_m (* im_m t_1) -0.16666666666666666) -1.0))
        (*
         (fma (* re re) -0.25 0.5)
         (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double t_1 = fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * (fma((re * re), fma(re, (re * 0.041666666666666664), -0.5), 1.0) * fma(t_1, (im_m * (im_m * (im_m * im_m))), -1.0));
	} else if (t_0 <= 0.0) {
		tmp = im_m * fma((im_m * im_m), fma(im_m, (im_m * t_1), -0.16666666666666666), -1.0);
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	t_1 = fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * Float64(fma(Float64(re * re), fma(re, Float64(re * 0.041666666666666664), -0.5), 1.0) * fma(t_1, Float64(im_m * Float64(im_m * Float64(im_m * im_m))), -1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * t_1), -0.16666666666666666), -1.0));
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * t$95$1), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $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

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 76.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. lower-*.f64 69.2

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.041666666666666664}, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   8. Simplified 69.2%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 69.2%

       \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \color{blue}{-1}\right)\right) \]

   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 6.2%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 52.1%

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

   9. Taylor expanded in im around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

   11. Simplified 52.1%

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

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), -1\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right)\right), im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 -2e-5)
      (*
       im_m
       (*
        (fma (* re re) (fma re (* re 0.041666666666666664) -0.5) 1.0)
        (fma
         (*
          (* im_m im_m)
          (*
           im_m
           (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)))
         im_m
         (fma im_m (* im_m -0.16666666666666666) -1.0))))
      (if (<= t_0 0.0)
        (- im_m)
        (*
         (fma (* re re) -0.25 0.5)
         (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -2e-5) {
		tmp = im_m * (fma((re * re), fma(re, (re * 0.041666666666666664), -0.5), 1.0) * fma(((im_m * im_m) * (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333))), im_m, fma(im_m, (im_m * -0.16666666666666666), -1.0)));
	} else if (t_0 <= 0.0) {
		tmp = -im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= -2e-5)
		tmp = Float64(im_m * Float64(fma(Float64(re * re), fma(re, Float64(re * 0.041666666666666664), -0.5), 1.0) * fma(Float64(Float64(im_m * im_m) * Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333))), im_m, fma(im_m, Float64(im_m * -0.16666666666666666), -1.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e-5], N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * N[(re * N[(re * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * im$95$m + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-im$95$m), N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $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))) < -2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 76.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \left(\color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{24} \cdot {re}^{2} - \frac{1}{2}\right) + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{24} \cdot {re}^{2} - \frac{1}{2}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{24} \cdot {re}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{{re}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{re \cdot \left(re \cdot \frac{1}{24}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, re \cdot \left(re \cdot \frac{1}{24}\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \color{blue}{\mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right)}, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. lower-*.f64 69.8

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, \color{blue}{re \cdot 0.041666666666666664}, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]

   8. Simplified 69.8%

      \[\leadsto im \cdot \left(\color{blue}{\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot \left(im \cdot \left(im \cdot im\right)\right), \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{5040} + \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right)} \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right) + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} + \left(im \cdot \left(im \cdot \frac{-1}{6}\right) + -1\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{6}\right)} + -1\right)\right)\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) + \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot \left(im \cdot im\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot im\right)} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot im} + \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot \frac{1}{24}, \frac{-1}{2}\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, \frac{-1}{5040}, \frac{-1}{120}\right) \cdot im\right) \cdot \left(im \cdot im\right), im, \mathsf{fma}\left(im, im \cdot \frac{-1}{6}, -1\right)\right)}\right) \]

   10. Applied egg-rr 69.8%

       \[\leadsto im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right)\right) \cdot \left(im \cdot im\right), im, \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)}\right) \]

   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))) < 0.0

   1. Initial program 5.7%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in re around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 51.7

         \[\leadsto \color{blue}{-im} \]

   8. Simplified 51.7%

      \[\leadsto \color{blue}{-im} \]

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -2 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(re, re \cdot 0.041666666666666664, -0.5\right), 1\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right)\right), im, \mathsf{fma}\left(im, im \cdot -0.16666666666666666, -1\right)\right)\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, \mathsf{fma}\left(im\_m, \mathsf{fma}\left(im\_m, -0.020833333333333332, -0.08333333333333333\right), -0.25\right), -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (fma
        im_m
        (fma im_m (fma im_m -0.020833333333333332 -0.08333333333333333) -0.25)
        -0.5))
      (if (<= t_0 0.0)
        (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0))
        (* im_m (fma 0.5 (* re re) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (exp(-im_m) - exp(im_m)) * (0.5 * cos(re));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * fma(im_m, fma(im_m, fma(im_m, -0.020833333333333332, -0.08333333333333333), -0.25), -0.5);
	} else if (t_0 <= 0.0) {
		tmp = im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0);
	} else {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * fma(im_m, fma(im_m, fma(im_m, -0.020833333333333332, -0.08333333333333333), -0.25), -0.5));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0));
	else
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.020833333333333332 + -0.08333333333333333), $MachinePrecision] + -0.25), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) - \frac{1}{4}\right) - \frac{1}{2}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) - \frac{1}{4}\right) - \frac{1}{2}\right)} \]

       2. sub-neg N/A

          \[\leadsto im \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) - \frac{1}{4}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

       3. metadata-eval N/A

          \[\leadsto im \cdot \left(im \cdot \left(im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) - \frac{1}{4}\right) + \color{blue}{\frac{-1}{2}}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) - \frac{1}{4}, \frac{-1}{2}\right)} \]

       5. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{-1}{2}\right) \]

       6. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, im \cdot \left(\frac{-1}{48} \cdot im - \frac{1}{12}\right) + \color{blue}{\frac{-1}{4}}, \frac{-1}{2}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, \frac{-1}{48} \cdot im - \frac{1}{12}, \frac{-1}{4}\right)}, \frac{-1}{2}\right) \]

       8. sub-neg N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\frac{-1}{48} \cdot im + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{-1}{4}\right), \frac{-1}{2}\right) \]

       9. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{im \cdot \frac{-1}{48}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{-1}{4}\right), \frac{-1}{2}\right) \]

       10. metadata-eval N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, im \cdot \frac{-1}{48} + \color{blue}{\frac{-1}{12}}, \frac{-1}{4}\right), \frac{-1}{2}\right) \]

       11. lower-fma.f64 75.5

           \[\leadsto im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \color{blue}{\mathsf{fma}\left(im, -0.020833333333333332, -0.08333333333333333\right)}, -0.25\right), -0.5\right) \]

   11. Simplified 75.5%

       \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \mathsf{fma}\left(im, -0.020833333333333332, -0.08333333333333333\right), -0.25\right), -0.5\right)} \]

   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 6.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 6.2

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

   5. Simplified 6.2%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 52.1

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

   8. Simplified 52.1%

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

   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 97.5%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 9.0

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

   5. Simplified 9.0%

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

   6. Taylor expanded in re around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 25.3

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

   8. Simplified 25.3%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, \mathsf{fma}\left(im, \mathsf{fma}\left(im, -0.020833333333333332, -0.08333333333333333\right), -0.25\right), -0.5\right)\\ \mathbf{elif}\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right) \leq 0:\\ \;\;\;\;im \cdot \mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   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)))

   1. Initial program 38.3%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 96.1%

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

4. Final simplification 92.6%

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

Alternative 9: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   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)))

   1. Initial program 38.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 96.1

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

   5. Simplified 96.1%

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

4. Final simplification 92.6%

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

Alternative 10: 53.2% accurate, 0.9× speedup

Math

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

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) (- INFINITY))
    (* im_m (* im_m (* im_m -0.16666666666666666)))
    (- im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= -((double) INFINITY)) {
		tmp = im_m * (im_m * (im_m * -0.16666666666666666));
	} else {
		tmp = -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 tmp;
	if (((Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * Math.cos(re))) <= -Double.POSITIVE_INFINITY) {
		tmp = im_m * (im_m * (im_m * -0.16666666666666666));
	} 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.exp(-im_m) - math.exp(im_m)) * (0.5 * math.cos(re))) <= -math.inf:
		tmp = im_m * (im_m * (im_m * -0.16666666666666666))
	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 (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= Float64(-Inf))
		tmp = Float64(im_m * Float64(im_m * Float64(im_m * -0.16666666666666666)));
	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 (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= -Inf)
		tmp = im_m * (im_m * (im_m * -0.16666666666666666));
	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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im$95$m)]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.8

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

   8. Simplified 48.8%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {im}^{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{im}^{3} \cdot \frac{-1}{6}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 48.8

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

   11. Simplified 48.8%

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

   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 38.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in re around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 36.0

         \[\leadsto \color{blue}{-im} \]

   8. Simplified 36.0%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.5%

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

Alternative 11: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot \cos re\right) \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, -0.25, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

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 (<= (* (- (exp (- im_m)) (exp im_m)) (* 0.5 (cos re))) (- INFINITY))
    (* im_m (fma im_m -0.25 -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 (((exp(-im_m) - exp(im_m)) * (0.5 * cos(re))) <= -((double) INFINITY)) {
		tmp = im_m * fma(im_m, -0.25, -0.5);
	} 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 (Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * cos(re))) <= Float64(-Inf))
		tmp = Float64(im_m * fma(im_m, -0.25, -0.5));
	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[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(im$95$m * N[(im$95$m * -0.25 + -0.5), $MachinePrecision]), $MachinePrecision], (-im$95$m)]), $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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 78.4

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

   5. Simplified 78.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   9. Taylor expanded in im around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{4} \cdot im - \frac{1}{2}\right)} \]

       2. sub-neg N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 44.4

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

   11. Simplified 44.4%

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

   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 38.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in re around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 36.0

         \[\leadsto \color{blue}{-im} \]

   8. Simplified 36.0%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

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

Alternative 12: 70.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), -0.16666666666666666\right), -1\right)\\ \end{array} \end{array} \]

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) -0.05)
    (*
     (fma (* re re) -0.25 0.5)
     (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0)))
    (*
     im_m
     (fma
      (* im_m im_m)
      (fma
       im_m
       (*
        im_m
        (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333))
       -0.16666666666666666)
      -1.0)))))

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) <= -0.05) {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	} else {
		tmp = im_m * fma((im_m * im_m), fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0);
	}
	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) <= -0.05)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	else
		tmp = Float64(im_m * fma(Float64(im_m * im_m), fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333)), -0.16666666666666666), -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 45.1%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.1

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

   8. Simplified 44.1%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 52.7%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

   5. Simplified 90.7%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 77.5%

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

   9. Taylor expanded in im around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-fma.f64 N/A

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

   11. Simplified 77.5%

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

Alternative 13: 68.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.3333333333333333, -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \end{array} \end{array} \]

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) -0.05)
    (*
     (fma (* re re) -0.25 0.5)
     (* im_m (fma (* im_m im_m) -0.3333333333333333 -2.0)))
    (*
     im_m
     (fma
      im_m
      (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
      -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma((re * re), -0.25, 0.5) * (im_m * fma((im_m * im_m), -0.3333333333333333, -2.0));
	} else {
		tmp = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(im_m * fma(Float64(im_m * im_m), -0.3333333333333333, -2.0)));
	else
		tmp = Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 45.1%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.1

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

   8. Simplified 44.1%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 52.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 76.4

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

   8. Simplified 76.4%

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

Alternative 14: 66.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right), -1\right)\\ \end{array} \end{array} \]

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) -0.05)
    (* im_m (fma 0.5 (* re re) -1.0))
    (*
     im_m
     (fma
      im_m
      (* im_m (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666))
      -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	} else {
		tmp = im_m * fma(im_m, (im_m * fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	else
		tmp = Float64(im_m * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666)), -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 45.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 60.2

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

   5. Simplified 60.2%

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

   6. Taylor expanded in re around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 32.6

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

   8. Simplified 32.6%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 52.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 76.4

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

   8. Simplified 76.4%

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

Alternative 15: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(0.5, re \cdot re, -1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\\ \end{array} \end{array} \]

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) -0.05)
    (* im_m (fma 0.5 (* re re) -1.0))
    (* im_m (fma -0.16666666666666666 (* im_m im_m) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = im_m * fma(0.5, (re * re), -1.0);
	} else {
		tmp = im_m * fma(-0.16666666666666666, (im_m * im_m), -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(im_m * fma(0.5, Float64(re * re), -1.0));
	else
		tmp = Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 45.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 60.2

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

   5. Simplified 60.2%

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

   6. Taylor expanded in re around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 32.6

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

   8. Simplified 32.6%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 52.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 52.5

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

   5. Simplified 52.5%

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

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.3

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

   8. Simplified 68.3%

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

Alternative 16: 53.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right)\right) \end{array} \]

FPCore

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

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * fma(-0.16666666666666666, Float64(im_m * im_m), -1.0)))
end

Wolfram

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

Derivation

1. Initial program 50.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-exp.f64 38.7

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

5. Simplified 38.7%

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

6. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 49.8

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

8. Simplified 49.8%

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

Alternative 17: 29.9% 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 50.6%

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

3. Taylor expanded in im around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 55.2

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

5. Simplified 55.2%

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

6. Taylor expanded in re around 0

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

   1. neg-mul-1 N/A

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

   2. lower-neg.f64 29.6

      \[\leadsto \color{blue}{-im} \]

8. Simplified 29.6%

   \[\leadsto \color{blue}{-im} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

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 2024207 
(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))))