math.cos on complex, imaginary part

Percentage Accurate: 66.3% → 99.7%

Time: 8.0s

Alternatives: 14

Speedup: 2.8×

• 49.9% 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: 66.3% 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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -4.0)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* (pow im_m 3.0) -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 <= -4.0) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -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 <= (-4.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-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 <= -4.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -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 <= -4.0:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -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 <= -4.0)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -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 <= -4.0)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -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, -4.0], 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] * -0.16666666666666666), $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)) < -4

   1. Initial program 100.0%

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

   if -4 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 55.5%

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

   2. Taylor expanded in im around 0 87.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 87.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 87.4%

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

      3. unsub-neg 87.4%

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

      4. associate-*r* 87.4%

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

      5. distribute-rgt-out-- 87.4%

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

      6. *-commutative 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -4:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 95.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 5600:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \mathbf{elif}\;im_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im_m} - e^{im_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5600.0)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 5.6e+102)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5600.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 5600.0d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 5.6d+102) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (im_m ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    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 <= 5600.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 5600.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 5.6e+102:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5600.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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 <= 5600.0)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 5.6e+102)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = (im_m ^ 3.0) * (sin(re) * -0.16666666666666666);
	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, 5600.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], 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, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5600

   1. Initial program 56.0%

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

   2. Taylor expanded in im around 0 86.6%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 86.6%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. associate-*r* 86.6%

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

      5. distribute-rgt-out-- 86.6%

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

      6. *-commutative 86.6%

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

   4. Simplified 86.6%

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

   if 5600 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 69.2%

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

      1. associate-*r* 69.2%

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

   4. Simplified 69.2%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 440:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{elif}\;im_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 440.0)
    (* (- im_m) (sin re))
    (if (<= im_m 5.6e+102)
      (* (- im_m) (pow (sin re) -3.0))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 440.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 5.6d+102) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = (im_m ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    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 <= 440.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 440.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 5.6e+102:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 440.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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 <= 440.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 5.6e+102)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = (im_m ^ 3.0) * (sin(re) * -0.16666666666666666);
	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, 440.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 440

   1. Initial program 55.7%

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

   2. Taylor expanded in im around 0 64.3%

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

      1. associate-*r* 64.3%

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

      2. neg-mul-1 64.3%

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

   4. Simplified 64.3%

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

   if 440 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   4. Simplified 3.1%

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

   6. Applied egg-rr 33.1%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 5600:\\ \;\;\;\;\sin re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \mathbf{elif}\;im_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5600.0)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 5.6e+102)
      (* (- im_m) (pow (sin re) -3.0))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5600.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 5600.0d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 5.6d+102) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = (im_m ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    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 <= 5600.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 5600.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 5.6e+102:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5600.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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 <= 5600.0)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 5.6e+102)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = (im_m ^ 3.0) * (sin(re) * -0.16666666666666666);
	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, 5600.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5600

   1. Initial program 56.0%

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

   2. Taylor expanded in im around 0 86.6%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 86.6%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 86.6%

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

      3. unsub-neg 86.6%

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

      4. associate-*r* 86.6%

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

      5. distribute-rgt-out-- 86.6%

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

      6. *-commutative 86.6%

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

   4. Simplified 86.6%

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

   if 5600 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   4. Simplified 3.1%

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

   6. Applied egg-rr 35.4%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5600:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 5: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 450:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{elif}\;im_m \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 450.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.7e+104)
      (* (- im_m) (pow (sin re) -3.0))
      (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 450.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = re * ((pow(im_m, 3.0) * -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) :: tmp
    if (im_m <= 450.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.7d+104) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = re * (((im_m ** 3.0d0) * (-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 tmp;
	if (im_m <= 450.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -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):
	tmp = 0
	if im_m <= 450.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.7e+104:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 450.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.7e+104)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -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)
	tmp = 0.0;
	if (im_m <= 450.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.7e+104)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = re * (((im_m ^ 3.0) * -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_] := N[(im$95$s * If[LessEqual[im$95$m, 450.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.7e+104], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 450

   1. Initial program 55.7%

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

   2. Taylor expanded in im around 0 64.3%

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

      1. associate-*r* 64.3%

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

      2. neg-mul-1 64.3%

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

   4. Simplified 64.3%

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

   if 450 < im < 1.6999999999999998e104

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   4. Simplified 3.1%

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

   6. Applied egg-rr 33.1%

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

   if 1.6999999999999998e104 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 80.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 230000000:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{elif}\;im_m \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im_m\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 230000000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.7e+104)
      (* (- im_m) (pow re -3.0))
      (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 230000000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * pow(re, -3.0);
	} else {
		tmp = re * ((pow(im_m, 3.0) * -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) :: tmp
    if (im_m <= 230000000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.7d+104) then
        tmp = -im_m * (re ** (-3.0d0))
    else
        tmp = re * (((im_m ** 3.0d0) * (-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 tmp;
	if (im_m <= 230000000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * Math.pow(re, -3.0);
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -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):
	tmp = 0
	if im_m <= 230000000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.7e+104:
		tmp = -im_m * math.pow(re, -3.0)
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 230000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.7e+104)
		tmp = Float64(Float64(-im_m) * (re ^ -3.0));
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -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)
	tmp = 0.0;
	if (im_m <= 230000000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.7e+104)
		tmp = -im_m * (re ^ -3.0);
	else
		tmp = re * (((im_m ^ 3.0) * -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_] := N[(im$95$s * If[LessEqual[im$95$m, 230000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.7e+104], N[((-im$95$m) * N[Power[re, -3.0], $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e8

   1. Initial program 56.4%

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

   2. Taylor expanded in im around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. neg-mul-1 63.4%

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

   4. Simplified 63.4%

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

   if 2.3e8 < im < 1.6999999999999998e104

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   4. Simplified 3.1%

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

   6. Applied egg-rr 41.2%

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

   7. Taylor expanded in re around 0 40.6%

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

   if 1.6999999999999998e104 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.8%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 230000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 7: 66.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 230000000:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{elif}\;im_m \leq 1.7 \cdot 10^{+162} \lor \neg \left(im_m \leq 6.4 \cdot 10^{+234}\right):\\ \;\;\;\;\left(-im_m\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(-im_m\right) \cdot \mathsf{expm1}\left(re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 230000000.0)
    (* (- im_m) (sin re))
    (if (or (<= im_m 1.7e+162) (not (<= im_m 6.4e+234)))
      (* (- im_m) (pow re -3.0))
      (* (- im_m) (expm1 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 <= 230000000.0) {
		tmp = -im_m * sin(re);
	} else if ((im_m <= 1.7e+162) || !(im_m <= 6.4e+234)) {
		tmp = -im_m * pow(re, -3.0);
	} else {
		tmp = -im_m * expm1(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 (im_m <= 230000000.0) {
		tmp = -im_m * Math.sin(re);
	} else if ((im_m <= 1.7e+162) || !(im_m <= 6.4e+234)) {
		tmp = -im_m * Math.pow(re, -3.0);
	} else {
		tmp = -im_m * Math.expm1(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 <= 230000000.0:
		tmp = -im_m * math.sin(re)
	elif (im_m <= 1.7e+162) or not (im_m <= 6.4e+234):
		tmp = -im_m * math.pow(re, -3.0)
	else:
		tmp = -im_m * math.expm1(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 <= 230000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif ((im_m <= 1.7e+162) || !(im_m <= 6.4e+234))
		tmp = Float64(Float64(-im_m) * (re ^ -3.0));
	else
		tmp = Float64(Float64(-im_m) * expm1(re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 230000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 1.7e+162], N[Not[LessEqual[im$95$m, 6.4e+234]], $MachinePrecision]], N[((-im$95$m) * N[Power[re, -3.0], $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e8

   1. Initial program 56.4%

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

   2. Taylor expanded in im around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. neg-mul-1 63.4%

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

   4. Simplified 63.4%

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

   if 2.3e8 < im < 1.70000000000000001e162 or 6.39999999999999983e234 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

   4. Simplified 4.9%

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in re around 0 44.0%

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

   if 1.70000000000000001e162 < im < 6.39999999999999983e234

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 5.0%

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

      1. associate-*r* 5.0%

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

      2. neg-mul-1 5.0%

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

   4. Simplified 5.0%

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

   6. Applied egg-rr 5.0%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

   7. Taylor expanded in re around 0 22.6%

      \[\leadsto \left(-im\right) \cdot \mathsf{expm1}\left(\color{blue}{re}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 230000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+162} \lor \neg \left(im \leq 6.4 \cdot 10^{+234}\right):\\ \;\;\;\;\left(-im\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \mathsf{expm1}\left(re\right)\\ \end{array} \]

Alternative 8: 80.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 230000000:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{elif}\;im_m \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im_m\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 230000000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.7e+104)
      (* (- im_m) (pow re -3.0))
      (* -0.16666666666666666 (* re (pow 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) {
	double tmp;
	if (im_m <= 230000000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * pow(re, -3.0);
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	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 <= 230000000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.7d+104) then
        tmp = -im_m * (re ** (-3.0d0))
    else
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    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 <= 230000000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.7e+104) {
		tmp = -im_m * Math.pow(re, -3.0);
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 230000000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.7e+104:
		tmp = -im_m * math.pow(re, -3.0)
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	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 (im_m <= 230000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.7e+104)
		tmp = Float64(Float64(-im_m) * (re ^ -3.0));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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)
	tmp = 0.0;
	if (im_m <= 230000000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.7e+104)
		tmp = -im_m * (re ^ -3.0);
	else
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	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_] := N[(im$95$s * If[LessEqual[im$95$m, 230000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.7e+104], N[((-im$95$m) * N[Power[re, -3.0], $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.3e8

   1. Initial program 56.4%

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

   2. Taylor expanded in im around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. neg-mul-1 63.4%

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

   4. Simplified 63.4%

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

   if 2.3e8 < im < 1.6999999999999998e104

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   4. Simplified 3.1%

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

   6. Applied egg-rr 41.2%

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

   7. Taylor expanded in re around 0 40.6%

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

   if 1.6999999999999998e104 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 73.8%

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

   6. Taylor expanded in im around inf 73.8%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 230000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\left(-im\right) \cdot {re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 9: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 38000:\\ \;\;\;\;\left(-im_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(-im_m\right) \cdot \mathsf{expm1}\left(re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 38000.0) (* (- im_m) (sin re)) (* (- im_m) (expm1 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 <= 38000.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -im_m * expm1(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 (im_m <= 38000.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -im_m * Math.expm1(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 <= 38000.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -im_m * math.expm1(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 <= 38000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(-im_m) * expm1(re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 38000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 38000

   1. Initial program 56.0%

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

   2. Taylor expanded in im around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. neg-mul-1 64.0%

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

   4. Simplified 64.0%

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

   if 38000 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

   4. Simplified 4.9%

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

   6. Applied egg-rr 4.9%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

   7. Taylor expanded in re around 0 13.5%

      \[\leadsto \left(-im\right) \cdot \mathsf{expm1}\left(\color{blue}{re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 38000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \mathsf{expm1}\left(re\right)\\ \end{array} \]

Alternative 10: 33.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \left(\left(-im_m\right) \cdot \mathsf{expm1}\left(re\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (- im_m) (expm1 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 * expm1(re));
}

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 * Math.expm1(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 * math.expm1(re))

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(-im_m) * expm1(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) * N[(Exp[re] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in im around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. neg-mul-1 51.3%

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

4. Simplified 51.3%

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

6. Applied egg-rr 51.3%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)} \]

7. Taylor expanded in re around 0 32.8%

   \[\leadsto \left(-im\right) \cdot \mathsf{expm1}\left(\color{blue}{re}\right) \]

8. Final simplification 32.8%

   \[\leadsto \left(-im\right) \cdot \mathsf{expm1}\left(re\right) \]

Alternative 11: 33.6% accurate, 77.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m (- re))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in im around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. neg-mul-1 51.3%

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

4. Simplified 51.3%

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

5. Taylor expanded in re around 0 33.0%

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

   1. associate-*r* 33.0%

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

   2. neg-mul-1 33.0%

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

7. Simplified 33.0%

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

8. Final simplification 33.0%

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

Alternative 12: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot -3 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -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 * -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 * (-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 * -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 * -3.0

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -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 * -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 * -3.0), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in im around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. neg-mul-1 51.3%

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

4. Simplified 51.3%

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

6. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.6%

   \[\leadsto -3 \]

Alternative 13: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot -0.004629629629629629 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -0.004629629629629629))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -0.004629629629629629;
}

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 * (-0.004629629629629629d0)
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 * -0.004629629629629629;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -0.004629629629629629

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -0.004629629629629629)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -0.004629629629629629;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -0.004629629629629629), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in im around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. neg-mul-1 51.3%

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

4. Simplified 51.3%

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

6. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-0.004629629629629629} \]

7. Final simplification 2.6%

   \[\leadsto -0.004629629629629629 \]

Alternative 14: 15.3% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * 0.0d0
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in im around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. neg-mul-1 51.3%

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

4. Simplified 51.3%

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

6. Applied egg-rr 14.4%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 14.4%

   \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× 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 2023319 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (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))))

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