math.cos on complex, imaginary part

Percentage Accurate: 65.9% → 99.6%

Time: 9.9s

Alternatives: 10

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	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 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 96.5%

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

4. Final simplification 97.2%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot \left({im\_m}^{2} \cdot -0.008333333333333333 - 0.16666666666666666\right) - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.04)
      (* t_0 (* 0.5 (sin re)))
      (*
       (sin re)
       (-
        (*
         (pow im_m 3.0)
         (- (* (pow im_m 2.0) -0.008333333333333333) 0.16666666666666666))
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * ((pow(im_m, 2.0) * -0.008333333333333333) - 0.16666666666666666)) - 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 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.04d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (((im_m ** 2.0d0) * (-0.008333333333333333d0)) - 0.16666666666666666d0)) - 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 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * ((Math.pow(im_m, 2.0) * -0.008333333333333333) - 0.16666666666666666)) - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * ((math.pow(im_m, 2.0) * -0.008333333333333333) - 0.16666666666666666)) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * Float64(Float64((im_m ^ 2.0) * -0.008333333333333333) - 0.16666666666666666)) - 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 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * (((im_m ^ 2.0) * -0.008333333333333333) - 0.16666666666666666)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.04], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0400000000000000008

   1. Initial program 99.9%

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

   if -0.0400000000000000008 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 53.4%

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

   3. Taylor expanded in im around 0 90.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.9%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.9%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 90.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 90.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 90.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 94.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in re around inf 94.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.04:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot \left({im}^{2} \cdot -0.008333333333333333 - 0.16666666666666666\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666670524691357 - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.005)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* (pow im_m 3.0) -0.16666670524691357) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666670524691357) - 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 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.005d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666670524691357d0)) - 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 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666670524691357) - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.005:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666670524691357) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666670524691357) - 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 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.005)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666670524691357) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.005], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666670524691357), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0050000000000000001

   1. Initial program 99.9%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 53.4%

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

   3. Taylor expanded in im around 0 90.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.9%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.9%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 90.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 90.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 90.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 94.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in re around inf 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) - im\right)} \]

   8. Applied egg-rr 87.5%

      \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot \color{blue}{4.6296296296296296 \cdot 10^{-6}} - 0.16666666666666666\right) - im\right) \]
   9. Step-by-step derivation

      1. metadata-eval 87.5%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(\color{blue}{-3.858024691358025 \cdot 10^{-8}} - 0.16666666666666666\right) - im\right) \]

      2. metadata-eval 87.5%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]

   10. Applied egg-rr 87.5%

       \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.005:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666670524691357 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -\infty:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) (- INFINITY))
    (* 8.0 (- 27.0 (exp im_m)))
    (* im_m (- (sin re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -((double) INFINITY)) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = im_m * -sin(re);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = im_m * -Math.sin(re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -math.inf:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = im_m * -math.sin(re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= Float64(-Inf))
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(im_m * Float64(-sin(re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -Inf)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = im_m * -sin(re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 45.3%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

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

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 70.2%

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

      1. associate-*r* 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.021:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666670524691357 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 10^{+61}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

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 (<= im_m 0.021)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666670524691357) im_m))
    (if (<= im_m 1e+61)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (pow im_m 5.0) (* (sin re) -0.008333333333333333))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.021) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666670524691357) - im_m);
	} else if (im_m <= 1e+61) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = pow(im_m, 5.0) * (sin(re) * -0.008333333333333333);
	}
	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 (im_m <= 0.021d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666670524691357d0)) - im_m)
    else if (im_m <= 1d+61) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (im_m ** 5.0d0) * (sin(re) * (-0.008333333333333333d0))
    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 (im_m <= 0.021) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666670524691357) - im_m);
	} else if (im_m <= 1e+61) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.pow(im_m, 5.0) * (Math.sin(re) * -0.008333333333333333);
	}
	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 im_m <= 0.021:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666670524691357) - im_m)
	elif im_m <= 1e+61:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.pow(im_m, 5.0) * (math.sin(re) * -0.008333333333333333)
	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 (im_m <= 0.021)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666670524691357) - im_m));
	elseif (im_m <= 1e+61)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64((im_m ^ 5.0) * Float64(sin(re) * -0.008333333333333333));
	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 (im_m <= 0.021)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666670524691357) - im_m);
	elseif (im_m <= 1e+61)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = (im_m ^ 5.0) * (sin(re) * -0.008333333333333333);
	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[im$95$m, 0.021], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666670524691357), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1e+61], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 5.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0210000000000000013

   1. Initial program 53.4%

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

   3. Taylor expanded in im around 0 90.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.9%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.9%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 90.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 90.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 90.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 94.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in re around inf 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) - im\right)} \]

   8. Applied egg-rr 87.5%

      \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot \color{blue}{4.6296296296296296 \cdot 10^{-6}} - 0.16666666666666666\right) - im\right) \]
   9. Step-by-step derivation

      1. metadata-eval 87.5%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(\color{blue}{-3.858024691358025 \cdot 10^{-8}} - 0.16666666666666666\right) - im\right) \]

      2. metadata-eval 87.5%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]

   10. Applied egg-rr 87.5%

       \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]

   if 0.0210000000000000013 < im < 9.99999999999999949e60

   1. Initial program 99.5%

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

   3. Taylor expanded in re around 0 87.0%

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

   if 9.99999999999999949e60 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 91.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 91.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. associate-*r* 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.021:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666670524691357 - im\right)\\ \mathbf{elif}\;im \leq 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 140:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666670524691357 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

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 (<= im_m 140.0)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666670524691357) im_m))
    (if (<= im_m 4.5e+61)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (pow im_m 5.0) (* (sin re) -0.008333333333333333))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 140.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666670524691357) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = pow(im_m, 5.0) * (sin(re) * -0.008333333333333333);
	}
	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 (im_m <= 140.0d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666670524691357d0)) - im_m)
    else if (im_m <= 4.5d+61) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = (im_m ** 5.0d0) * (sin(re) * (-0.008333333333333333d0))
    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 (im_m <= 140.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666670524691357) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.pow(im_m, 5.0) * (Math.sin(re) * -0.008333333333333333);
	}
	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 im_m <= 140.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666670524691357) - im_m)
	elif im_m <= 4.5e+61:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.pow(im_m, 5.0) * (math.sin(re) * -0.008333333333333333)
	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 (im_m <= 140.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666670524691357) - im_m));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64((im_m ^ 5.0) * Float64(sin(re) * -0.008333333333333333));
	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 (im_m <= 140.0)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666670524691357) - im_m);
	elseif (im_m <= 4.5e+61)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = (im_m ^ 5.0) * (sin(re) * -0.008333333333333333);
	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[im$95$m, 140.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666670524691357), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 5.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 140

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 90.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.9%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.9%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.9%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 90.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 90.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 90.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 90.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 94.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in re around inf 94.1%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) - im\right)} \]

   8. Applied egg-rr 87.4%

      \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot \color{blue}{4.6296296296296296 \cdot 10^{-6}} - 0.16666666666666666\right) - im\right) \]
   9. Step-by-step derivation

      1. metadata-eval 87.4%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \left(\color{blue}{-3.858024691358025 \cdot 10^{-8}} - 0.16666666666666666\right) - im\right) \]

      2. metadata-eval 87.4%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]

   10. Applied egg-rr 87.4%

       \[\leadsto \sin re \cdot \left({im}^{3} \cdot \color{blue}{-0.16666670524691357} - im\right) \]

   if 140 < im < 4.5e61

   1. Initial program 100.0%

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

   4. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 14.3%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 4.5e61 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 91.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 91.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. associate-*r* 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 280:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

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 (<= im_m 280.0)
    (* im_m (- (sin re)))
    (if (<= im_m 4.5e+61)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (pow im_m 5.0) (* (sin re) -0.008333333333333333))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 280.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = pow(im_m, 5.0) * (sin(re) * -0.008333333333333333);
	}
	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 (im_m <= 280.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 4.5d+61) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = (im_m ** 5.0d0) * (sin(re) * (-0.008333333333333333d0))
    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 (im_m <= 280.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 4.5e+61) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.pow(im_m, 5.0) * (Math.sin(re) * -0.008333333333333333);
	}
	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 im_m <= 280.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 4.5e+61:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.pow(im_m, 5.0) * (math.sin(re) * -0.008333333333333333)
	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 (im_m <= 280.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64((im_m ^ 5.0) * Float64(sin(re) * -0.008333333333333333));
	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 (im_m <= 280.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 4.5e+61)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = (im_m ^ 5.0) * (sin(re) * -0.008333333333333333);
	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[im$95$m, 280.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 5.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 280

   1. Initial program 53.6%

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

   3. Taylor expanded in im around 0 70.2%

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

      1. associate-*r* 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

   if 280 < im < 4.5e61

   1. Initial program 100.0%

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

   4. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 14.3%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 4.5e61 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 90.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 90.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) - \sin re\right)} \]

      4. distribute-lft-out-- 90.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - im \cdot \sin re} \]

      5. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - im \cdot \sin re \]

      6. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - im \cdot \sin re \]

      7. associate-*r* 91.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      8. distribute-rgt-out 91.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - im \cdot \sin re \]

      9. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \sin re\right) \cdot -0.008333333333333333} \]

      2. associate-*r* 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 280:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot \left(\sin re \cdot -0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.1 \cdot 10^{+76}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 3.1e+76) (* im_m (- (sin re))) (* (- im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.1e+76) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.1d+76) then
        tmp = im_m * -sin(re)
    else
        tmp = -im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.1e+76) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.1e+76:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.1e+76)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(-im_m) * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.1e+76)
		tmp = im_m * -sin(re);
	else
		tmp = -im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.1e+76], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.10000000000000011e76

   1. Initial program 55.8%

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

   3. Taylor expanded in im around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   if 3.10000000000000011e76 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around 0 9.9%

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

      1. associate-*r* 9.9%

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

      2. neg-mul-1 9.9%

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

   8. Simplified 9.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+76}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.005:\\ \;\;\;\;im\_m \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= (sin re) -0.005) (* im_m re) (* (- im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.005) {
		tmp = im_m * re;
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (sin(re) <= (-0.005d0)) then
        tmp = im_m * re
    else
        tmp = -im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (Math.sin(re) <= -0.005) {
		tmp = im_m * re;
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if math.sin(re) <= -0.005:
		tmp = im_m * re
	else:
		tmp = -im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.005)
		tmp = Float64(im_m * re);
	else
		tmp = Float64(Float64(-im_m) * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (sin(re) <= -0.005)
		tmp = im_m * re;
	else
		tmp = -im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.005], N[(im$95$m * re), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0050000000000000001

   1. Initial program 45.6%

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

   3. Taylor expanded in im around 0 62.2%

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

      1. associate-*r* 62.2%

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

      2. neg-mul-1 62.2%

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

   5. Simplified 62.2%

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

   6. Taylor expanded in re around 0 15.5%

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

      1. associate-*r* 15.5%

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

      2. neg-mul-1 15.5%

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

   8. Simplified 15.5%

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

      1. add-sqr-sqrt 11.1%

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

      2. sqrt-unprod 19.1%

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

      3. sqr-neg 19.1%

         \[\leadsto \sqrt{\color{blue}{im \cdot im}} \cdot re \]

      4. sqrt-prod 7.6%

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

      5. add-sqr-sqrt 12.3%

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

      6. pow1 12.3%

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

   10. Applied egg-rr 12.3%

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

       1. unpow1 12.3%

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

   12. Simplified 12.3%

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

   if -0.0050000000000000001 < (sin.f64 re)

   1. Initial program 69.0%

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

   3. Taylor expanded in im around 0 54.6%

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

      1. associate-*r* 54.6%

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

      2. neg-mul-1 54.6%

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

   5. Simplified 54.6%

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

   6. Taylor expanded in re around 0 39.7%

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

      1. associate-*r* 39.7%

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

      2. neg-mul-1 39.7%

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

   8. Simplified 39.7%

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

Alternative 10: 20.7% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * re)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * re);
}

Python

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

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * re);
end

Wolfram

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

Derivation

1. Initial program 63.2%

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

3. Taylor expanded in im around 0 56.5%

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

   1. associate-*r* 56.5%

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

   2. neg-mul-1 56.5%

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

5. Simplified 56.5%

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

6. Taylor expanded in re around 0 33.7%

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

   1. associate-*r* 33.7%

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

   2. neg-mul-1 33.7%

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

8. Simplified 33.7%

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

   1. add-sqr-sqrt 19.6%

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

   2. sqrt-unprod 34.9%

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

   3. sqr-neg 34.9%

      \[\leadsto \sqrt{\color{blue}{im \cdot im}} \cdot re \]

   4. sqrt-prod 10.0%

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

   5. add-sqr-sqrt 22.0%

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

   6. pow1 22.0%

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

10. Applied egg-rr 22.0%

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

    1. unpow1 22.0%

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

12. Simplified 22.0%

    \[\leadsto \color{blue}{im \cdot re} \]
13. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024180 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))