math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.7%

Time: 9.2s

Alternatives: 21

Speedup: 1.4×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= im_m 4.0)
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (* t_0 (- (- 1.0 im_m) (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (im_m <= 4.0) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * ((1.0 - im_m) - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if (im_m <= 4.0d0) then
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    else
        tmp = t_0 * ((1.0d0 - im_m) - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if (im_m <= 4.0) {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * ((1.0 - im_m) - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if im_m <= 4.0:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	else:
		tmp = t_0 * ((1.0 - im_m) - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (im_m <= 4.0)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 - im_m) - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if (im_m <= 4.0)
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	else
		tmp = t_0 * ((1.0 - im_m) - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if im < 4

   1. Initial program 37.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   if 4 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 - im\right)} - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \sinh \left(-im\_m\right)\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\left(t\_0 \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(3 \cdot \frac{2 \cdot t\_0}{3}\right)\\ \end{array} \end{array} \end{array} \]

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 (sinh (- im_m)))
        (t_1 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_1 -0.05)
      (* (* t_0 2.0) 0.5)
      (if (<= t_1 0.0002)
        (* (* (cos re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (* (fma (* re re) -0.25 0.5) (* 3.0 (/ (* 2.0 t_0) 3.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = sinh(-im_m);
	double t_1 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (t_0 * 2.0) * 0.5;
	} else if (t_1 <= 0.0002) {
		tmp = (cos(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = fma((re * re), -0.25, 0.5) * (3.0 * ((2.0 * t_0) / 3.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = sinh(Float64(-im_m))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(t_0 * 2.0) * 0.5);
	elseif (t_1 <= 0.0002)
		tmp = Float64(Float64(cos(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(3.0 * Float64(Float64(2.0 * t_0) / 3.0)));
	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[Sinh[(-im$95$m)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, -0.05], N[(N[(t$95$0 * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(3.0 * N[(N[(2.0 * t$95$0), $MachinePrecision] / 3.0), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 7.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right) \cdot \frac{2 \cdot \sinh \left(-im\right)}{\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right)}\right)} \]

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 19.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right) \cdot \frac{2 \cdot \sinh \left(-im\right)}{\color{blue}{3}}\right) \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} \cdot \frac{2 \cdot \sinh \left(-im\right)}{3}\right) \]

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 81.0

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \left(3 \cdot \frac{2 \cdot \sinh \left(-im\right)}{3}\right) \]

   13. Applied rewrites 81.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(3 \cdot \frac{2 \cdot \sinh \left(-im\right)}{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\_m\right) \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -0.05)
      (* (* (sinh (- im_m)) 2.0) 0.5)
      (if (<= t_0 0.0002)
        (* (* (cos re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (*
         (fma
          (-
           (*
            (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
            re)
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
              (* im_m im_m))
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -0.05) {
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	} else if (t_0 <= 0.0002) {
		tmp = (cos(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(sinh(Float64(-im_m)) * 2.0) * 0.5);
	elseif (t_0 <= 0.0002)
		tmp = Float64(Float64(cos(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.05], N[(N[(N[Sinh[(-im$95$m)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5 \]

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 7.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(\cos re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\left(\sinh \left(-im\_m\right) \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -5e-5)
      (* (* (sinh (- im_m)) 2.0) 0.5)
      (if (<= t_0 0.0002)
        (* (- (cos re)) im_m)
        (*
         (fma
          (-
           (*
            (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
            re)
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
              (* im_m im_m))
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	} else if (t_0 <= 0.0002) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -5e-5)
		tmp = Float64(Float64(sinh(Float64(-im_m)) * 2.0) * 0.5);
	elseif (t_0 <= 0.0002)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5e-5], N[(N[(N[Sinh[(-im$95$m)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 74.9

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

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.9%

      \[\leadsto \left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5 \]

   if -5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 6.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+108}:\\ \;\;\;\;\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -2e+108)
      (* (- (- 1.0 im_m) (exp im_m)) 0.5)
      (if (<= t_0 0.0002)
        (* (- (cos re)) im_m)
        (*
         (fma
          (-
           (*
            (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
            re)
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
              (* im_m im_m))
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -2e+108) {
		tmp = ((1.0 - im_m) - exp(im_m)) * 0.5;
	} else if (t_0 <= 0.0002) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -2e+108)
		tmp = Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * 0.5);
	elseif (t_0 <= 0.0002)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+108], N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -2.0000000000000001e108

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 75.7

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 75.7%

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

   if -2.0000000000000001e108 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 8.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -2 \cdot 10^{+108}:\\ \;\;\;\;\left(\left(1 - im\right) - e^{im}\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -5e-5)
      (*
       (fma (- (* 0.020833333333333332 (* re re)) 0.25) (* re re) 0.5)
       (*
        (-
         (*
          (-
           (*
            (*
             (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (if (<= t_0 0.0002)
        (* (- (cos re)) im_m)
        (*
         (fma
          (-
           (*
            (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
            re)
           0.25)
          (* re re)
          0.5)
         (*
          (-
           (*
            (-
             (*
              (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
              (* im_m im_m))
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = fma(((0.020833333333333332 * (re * re)) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else if (t_0 <= 0.0002) {
		tmp = -cos(re) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -5e-5)
		tmp = Float64(fma(Float64(Float64(0.020833333333333332 * Float64(re * re)) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	elseif (t_0 <= 0.0002)
		tmp = Float64(Float64(-cos(re)) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5e-5], N[(N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[((-N[Cos[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 70.4

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

   8. Applied rewrites 70.4%

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

   if -5.00000000000000024e-5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 2.0000000000000001e-4

   1. Initial program 6.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.2%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right) - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.0002:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 0.5× speedup

Math

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

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 (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (- (sqrt (* im_m im_m)))
      (if (<= t_0 0.0) (- im_m) (* 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 = (0.5 * cos(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -sqrt((im_m * im_m));
	} else if (t_0 <= 0.0) {
		tmp = -im_m;
	} 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(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(im_m * im_m)));
	elseif (t_0 <= 0.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(im_m * fma(Float64(0.5 * 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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], (-N[Sqrt[N[(im$95$m * im$95$m), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 0.0], (-im$95$m), N[(im$95$m * N[(N[(0.5 * re), $MachinePrecision] * re + -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 im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 51.9%

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

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

   1. Initial program 7.6%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 60.5%

      \[\leadsto -im \]

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

   1. Initial program 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.5%

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 16.5%

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

4. Final simplification 47.9%

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

Alternative 8: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot t\_1 \leq -0.05:\\ \;\;\;\;t\_1 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= (* t_0 t_1) -0.05)
      (* t_1 0.5)
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = exp(-im_m) - exp(im_m);
	double tmp;
	if ((t_0 * t_1) <= -0.05) {
		tmp = t_1 * 0.5;
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    t_1 = exp(-im_m) - exp(im_m)
    if ((t_0 * t_1) <= (-0.05d0)) then
        tmp = t_1 * 0.5d0
    else
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if ((t_0 * t_1) <= -0.05) {
		tmp = t_1 * 0.5;
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.cos(re)
	t_1 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if (t_0 * t_1) <= -0.05:
		tmp = t_1 * 0.5
	else:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (Float64(t_0 * t_1) <= -0.05)
		tmp = Float64(t_1 * 0.5);
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * cos(re);
	t_1 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if ((t_0 * t_1) <= -0.05)
		tmp = t_1 * 0.5;
	else
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * t$95$1), $MachinePrecision], -0.05], N[(t$95$1 * 0.5), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -0.050000000000000003 < (*.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 35.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\_m\right) \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) -0.05)
      (* (* (sinh (- im_m)) 2.0) 0.5)
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -0.05) {
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if ((t_0 * (exp(-im_m) - exp(im_m))) <= (-0.05d0)) then
        tmp = (sinh(-im_m) * 2.0d0) * 0.5d0
    else
        tmp = t_0 * (((((((((-0.0003968253968253968d0) * (im_m * im_m)) - 0.016666666666666666d0) * im_m) * im_m) - 0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if ((t_0 * (Math.exp(-im_m) - Math.exp(im_m))) <= -0.05) {
		tmp = (Math.sinh(-im_m) * 2.0) * 0.5;
	} else {
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if (t_0 * (math.exp(-im_m) - math.exp(im_m))) <= -0.05:
		tmp = (math.sinh(-im_m) * 2.0) * 0.5
	else:
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -0.05)
		tmp = Float64(Float64(sinh(Float64(-im_m)) * 2.0) * 0.5);
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -0.05)
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	else
		tmp = t_0 * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[Sinh[(-im$95$m)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5 \]

   if -0.050000000000000003 < (*.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 35.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\_m\right) \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot im\_m\right) \cdot im\_m - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) -0.05)
      (* (* (sinh (- im_m)) 2.0) 0.5)
      (*
       t_0
       (*
        (-
         (*
          (*
           (- (* -0.016666666666666666 (* im_m im_m)) 0.3333333333333333)
           im_m)
          im_m)
         2.0)
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -0.05) {
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * cos(re)
    if ((t_0 * (exp(-im_m) - exp(im_m))) <= (-0.05d0)) then
        tmp = (sinh(-im_m) * 2.0d0) * 0.5d0
    else
        tmp = t_0 * (((((((-0.016666666666666666d0) * (im_m * im_m)) - 0.3333333333333333d0) * im_m) * im_m) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.cos(re);
	double tmp;
	if ((t_0 * (Math.exp(-im_m) - Math.exp(im_m))) <= -0.05) {
		tmp = (Math.sinh(-im_m) * 2.0) * 0.5;
	} else {
		tmp = t_0 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.cos(re)
	tmp = 0
	if (t_0 * (math.exp(-im_m) - math.exp(im_m))) <= -0.05:
		tmp = (math.sinh(-im_m) * 2.0) * 0.5
	else:
		tmp = t_0 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -0.05)
		tmp = Float64(Float64(sinh(Float64(-im_m)) * 2.0) * 0.5);
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * cos(re);
	tmp = 0.0;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -0.05)
		tmp = (sinh(-im_m) * 2.0) * 0.5;
	else
		tmp = t_0 * ((((((-0.016666666666666666 * (im_m * im_m)) - 0.3333333333333333) * im_m) * im_m) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[Sinh[(-im$95$m)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.0%

      \[\leadsto \left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5 \]

   if -0.050000000000000003 < (*.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 35.6%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -0.05:\\ \;\;\;\;\left(\sinh \left(-im\right) \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot im\_m\right) \cdot im\_m - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.0006944444444444445 \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right) - 0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (-
      (*
       (-
        (*
         (*
          (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
          im_m)
         im_m)
        0.16666666666666666)
       (* im_m im_m))
      1.0)
     im_m)
    (*
     (fma
      (- (* (* (* -0.0006944444444444445 (* re re)) re) re) 0.25)
      (* re re)
      0.5)
     (*
      (-
       (*
        (-
         (*
          (*
           (- (* -0.0003968253968253968 (* im_m im_m)) 0.016666666666666666)
           im_m)
          im_m)
         0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = fma(((((-0.0006944444444444445 * (re * re)) * re) * re) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-0.0006944444444444445 * Float64(re * re)) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im_m * im_m)) - 0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.7%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 75.8

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 75.8%

       \[\leadsto \mathsf{fma}\left(\left(\left(-0.0006944444444444445 \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.0006944444444444445 \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot im\_m\right) \cdot im\_m - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (-
      (*
       (-
        (*
         (*
          (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
          im_m)
         im_m)
        0.16666666666666666)
       (* im_m im_m))
      1.0)
     im_m)
    (*
     (fma
      (-
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       0.25)
      (* re re)
      0.5)
     (*
      (-
       (*
        (-
         (*
          (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
          (* im_m im_m))
         0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * (im_m * im_m)) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * Float64(im_m * im_m)) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.7%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 75.8

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.8%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.8%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot im\_m\right) \cdot im\_m - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (-
      (*
       (-
        (*
         (*
          (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
          im_m)
         im_m)
        0.16666666666666666)
       (* im_m im_m))
      1.0)
     im_m)
    (*
     (fma
      (-
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       0.25)
      (* re re)
      0.5)
     (*
      (-
       (*
        (- (* (* -0.016666666666666666 im_m) im_m) 0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * ((((((-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.7%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\left(\frac{-1}{2520} \cdot \left(im \cdot im\right) - \frac{1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 75.8

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 68.3%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im\_m \cdot im\_m\right) - 0.008333333333333333\right) \cdot im\_m\right) \cdot im\_m - 0.16666666666666666\right) \cdot \left(im\_m \cdot im\_m\right) - 1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (-
      (*
       (-
        (*
         (*
          (- (* -0.0001984126984126984 (* im_m im_m)) 0.008333333333333333)
          im_m)
         im_m)
        0.16666666666666666)
       (* im_m im_m))
      1.0)
     im_m)
    (*
     (fma
      (-
       (*
        (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
        re)
       0.25)
      (* re re)
      0.5)
     (* (- (* -0.3333333333333333 (* im_m im_m)) 2.0) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.7%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

      15. lower-*.f64 60.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   8. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = ((((((((-0.0001984126984126984d0) * (im_m * im_m)) - 0.008333333333333333d0) * im_m) * im_m) - 0.16666666666666666d0) * (im_m * im_m)) - 1.0d0) * im_m
    else
        tmp = ((re * re) * (-0.25d0)) * ((((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	} else {
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * 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 ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m
	else:
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * Float64(im_m * im_m)) - 1.0) * im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * -0.25) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0) * 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 (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = (((((((-0.0001984126984126984 * (im_m * im_m)) - 0.008333333333333333) * im_m) * im_m) - 0.16666666666666666) * (im_m * im_m)) - 1.0) * im_m;
	else
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * 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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.7%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.4

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 18.4%

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

4. Final simplification 52.9%

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

Alternative 16: 70.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0d0) then
        tmp = ((((((-0.008333333333333333d0) * (im_m * im_m)) - 0.16666666666666666d0) * im_m) * im_m) - 1.0d0) * im_m
    else
        tmp = ((re * re) * (-0.25d0)) * ((((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= 0.0) {
		tmp = (((((-0.008333333333333333 * (im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m;
	} else {
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * 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 ((0.5 * math.cos(re)) * (math.exp(-im_m) - math.exp(im_m))) <= 0.0:
		tmp = (((((-0.008333333333333333 * (im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m
	else:
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * -0.25) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0) * 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 (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0)
		tmp = (((((-0.008333333333333333 * (im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m;
	else
		tmp = ((re * re) * -0.25) * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * 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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.1%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.4

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 18.4%

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

4. Final simplification 52.4%

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

Alternative 17: 68.5% accurate, 0.9× speedup

Math

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

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 (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (-
      (*
       (* (- (* -0.008333333333333333 (* im_m im_m)) 0.16666666666666666) im_m)
       im_m)
      1.0)
     im_m)
    (* 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 tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((((-0.008333333333333333 * (im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m;
	} 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)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im_m * im_m)) - 0.16666666666666666) * im_m) * im_m) - 1.0) * im_m);
	else
		tmp = Float64(im_m * fma(Float64(0.5 * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(im$95$m * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 63.1%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.5%

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 16.5%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot re, re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

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

Alternative 18: 63.9% accurate, 0.9× speedup

Math

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

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 (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (* (- (* (* im_m im_m) -0.16666666666666666) 1.0) im_m)
    (* 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 tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (((im_m * im_m) * -0.16666666666666666) - 1.0) * im_m;
	} 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)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) - 1.0) * im_m);
	else
		tmp = Float64(im_m * fma(Float64(0.5 * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(im$95$m * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 58.1%

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

   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 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.5%

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 16.5%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot re, re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

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

Alternative 19: 39.8% accurate, 0.9× speedup

Math

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

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 (<= (* (* 0.5 (cos re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (- im_m)
    (* 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 tmp;
	if (((0.5 * cos(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = -im_m;
	} 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)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(im_m * fma(Float64(0.5 * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], (-im$95$m), N[(im$95$m * N[(N[(0.5 * re), $MachinePrecision] * re + -1.0), $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))) < 0.0

   1. Initial program 37.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.1%

      \[\leadsto -im \]

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

   1. Initial program 98.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 7.5%

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

   10. Applied rewrites 0.3%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 16.5%

       \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot re, re, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

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

Alternative 20: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot \cos re\right) \cdot \left(3 \cdot \frac{2 \cdot \sinh \left(-im\_m\right)}{3}\right)\right) \end{array} \]

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 (* (* 0.5 (cos re)) (* 3.0 (/ (* 2.0 (sinh (- im_m))) 3.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * cos(re)) * (3.0 * ((2.0 * sinh(-im_m)) / 3.0)));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((0.5d0 * cos(re)) * (3.0d0 * ((2.0d0 * sinh(-im_m)) / 3.0d0)))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * Math.cos(re)) * (3.0 * ((2.0 * Math.sinh(-im_m)) / 3.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * math.cos(re)) * (3.0 * ((2.0 * math.sinh(-im_m)) / 3.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.5 * cos(re)) * Float64(3.0 * Float64(Float64(2.0 * sinh(Float64(-im_m))) / 3.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * cos(re)) * (3.0 * ((2.0 * sinh(-im_m)) / 3.0)));
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[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(3.0 * N[(N[(2.0 * N[Sinh[(-im$95$m)], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

4. Applied rewrites 52.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right) \cdot \frac{2 \cdot \sinh \left(-im\right)}{\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right)}\right)} \]

5. Taylor expanded in im around 0

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

7. Applied rewrites 74.6%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\mathsf{fma}\left(2 \cdot \cosh im, e^{im}, {\left(e^{im}\right)}^{-2}\right) \cdot \frac{2 \cdot \sinh \left(-im\right)}{\color{blue}{3}}\right) \]

8. Taylor expanded in im around 0

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

10. Applied rewrites 99.7%

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{3} \cdot \frac{2 \cdot \sinh \left(-im\right)}{3}\right) \]
11. Add Preprocessing

Alternative 21: 30.3% 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 52.2%

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

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cos.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 33.8%

   \[\leadsto -im \]
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 2024339 
(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))))