math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.2%

Time: 12.5s

Alternatives: 13

Speedup: 2.8×

• 49.3% 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: 65.5% 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: 99.2% accurate, 0.6× speedup

Math

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

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
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -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 * (((im_m * im_m) * (((im_m * im_m) * -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 * (((im_m * im_m) * (((im_m * im_m) * -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 * (((im_m * im_m) * (((im_m * im_m) * -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(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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 * im_m) * (((im_m * im_m) * -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[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $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 49.9%

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

   3. Taylor expanded in im around 0 93.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 93.9%

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

   5. Applied egg-rr 93.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 93.9%

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

   7. Applied egg-rr 93.9%

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

4. Final simplification 95.3%

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

Alternative 2: 94.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 500 \lor \neg \left(im\_m \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 500.0) (not (<= im_m 1.02e+62)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0)))
    (* 8.0 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 500.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 500.0d0) .or. (.not. (im_m <= 1.02d+62))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 500.0) || !(im_m <= 1.02e+62)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 500.0) or not (im_m <= 1.02e+62):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	else:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 500.0) || !(im_m <= 1.02e+62))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 500.0) || ~((im_m <= 1.02e+62)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0));
	else
		tmp = 8.0 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 500.0], N[Not[LessEqual[im$95$m, 1.02e+62]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 500 or 1.02000000000000002e62 < im

   1. Initial program 60.1%

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

   3. Taylor expanded in im around 0 95.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 95.1%

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

   5. Applied egg-rr 95.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 95.1%

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

   7. Applied egg-rr 95.1%

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

   if 500 < im < 1.02000000000000002e62

   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 70.0%

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

   6. Applied egg-rr 70.0%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500 \lor \neg \left(im \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;t\_0 \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \end{array} \]

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
         (*
          im_m
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
           2.0))))
   (* im_s (if (<= (sin re) -0.01) (* t_0 -2.0) (* t_0 (* 0.5 re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (sin(re) <= -0.01) {
		tmp = t_0 * -2.0;
	} else {
		tmp = t_0 * (0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)
    if (sin(re) <= (-0.01d0)) then
        tmp = t_0 * (-2.0d0)
    else
        tmp = t_0 * (0.5d0 * 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 t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	double tmp;
	if (Math.sin(re) <= -0.01) {
		tmp = t_0 * -2.0;
	} else {
		tmp = t_0 * (0.5 * 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):
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)
	tmp = 0
	if math.sin(re) <= -0.01:
		tmp = t_0 * -2.0
	else:
		tmp = t_0 * (0.5 * re)
	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(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))
	tmp = 0.0
	if (sin(re) <= -0.01)
		tmp = Float64(t_0 * -2.0);
	else
		tmp = Float64(t_0 * Float64(0.5 * 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)
	t_0 = im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0);
	tmp = 0.0;
	if (sin(re) <= -0.01)
		tmp = t_0 * -2.0;
	else
		tmp = t_0 * (0.5 * 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_] := Block[{t$95$0 = N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.01], N[(t$95$0 * -2.0), $MachinePrecision], N[(t$95$0 * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.0100000000000000002

   1. Initial program 47.6%

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

   3. Taylor expanded in im around 0 93.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 93.1%

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

   5. Applied egg-rr 93.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 93.1%

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

   7. Applied egg-rr 93.1%

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

   9. Applied egg-rr 45.8%

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

   if -0.0100000000000000002 < (sin.f64 re)

   1. Initial program 67.1%

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

   3. Taylor expanded in im around 0 91.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 91.0%

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

   5. Applied egg-rr 91.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 91.0%

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

   7. Applied egg-rr 91.0%

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

   8. Taylor expanded in re around 0 72.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.01:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 260:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 260.0) (* (- im_m) (sin re)) (* 8.0 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 260.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = 8.0 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 260.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 260.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 260.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 260.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 260.0)
		tmp = -im_m * sin(re);
	else
		tmp = 8.0 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 260.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 260

   1. Initial program 49.9%

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

   3. Taylor expanded in im around 0 72.7%

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

      1. associate-*r* 72.7%

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

      2. neg-mul-1 72.7%

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

   5. Simplified 72.7%

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

   if 260 < im

   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 53.3%

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

   6. Applied egg-rr 53.3%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

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 (<= im_m 2.55e+15)
    (* (- im_m) (sin re))
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     (* 0.5 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 <= 2.55e+15) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 2.55d+15) then
        tmp = -im_m * sin(re)
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * (0.5d0 * 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 <= 2.55e+15) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 2.55e+15:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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 <= 2.55e+15)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * Float64(0.5 * 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 <= 2.55e+15)
		tmp = -im_m * sin(re);
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * (0.5 * 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, 2.55e+15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.55e15

   1. Initial program 50.4%

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

   3. Taylor expanded in im around 0 71.9%

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

      1. associate-*r* 71.9%

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

      2. neg-mul-1 71.9%

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

   5. Simplified 71.9%

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

   if 2.55e15 < 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 86.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 86.9%

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

   5. Applied egg-rr 86.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 86.9%

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

   7. Applied egg-rr 86.9%

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

   8. Taylor expanded in re around 0 65.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.4 \cdot 10^{+58}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \end{array} \]

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 3.4e+58)
    (* im_m (- re))
    (*
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
       2.0))
     0.25))))

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 <= 3.4e+58) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 <= 3.4d+58) then
        tmp = im_m * -re
    else
        tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)) * 0.25d0
    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 <= 3.4e+58) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	}
	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 <= 3.4e+58:
		tmp = im_m * -re
	else:
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25
	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 <= 3.4e+58)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25);
	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 <= 3.4e+58)
		tmp = im_m * -re;
	else
		tmp = (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)) * 0.25;
	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, 3.4e+58], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.4000000000000001e58

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in re around 0 39.5%

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

      1. associate-*r* 39.5%

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

      2. mul-1-neg 39.5%

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

   8. Simplified 39.5%

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

   if 3.4000000000000001e58 < 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 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(-0.016666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Applied egg-rr 50.0%

      \[\leadsto \color{blue}{0.25} \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) - 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4 \cdot 10^{+58}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.55 \cdot 10^{+68}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 - im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \end{array} \]

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 1.55e+68)
    (* im_m (- re))
    (- (* im_m (+ 2.0 (* im_m (- 1.0 (* im_m 0.3333333333333333))))) 52.0))))

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 <= 1.55e+68) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (1.0 - (im_m * 0.3333333333333333))))) - 52.0;
	}
	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 <= 1.55d+68) then
        tmp = im_m * -re
    else
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 - (im_m * 0.3333333333333333d0))))) - 52.0d0
    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 <= 1.55e+68) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (1.0 - (im_m * 0.3333333333333333))))) - 52.0;
	}
	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 <= 1.55e+68:
		tmp = im_m * -re
	else:
		tmp = (im_m * (2.0 + (im_m * (1.0 - (im_m * 0.3333333333333333))))) - 52.0
	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 <= 1.55e+68)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 - Float64(im_m * 0.3333333333333333))))) - 52.0);
	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 <= 1.55e+68)
		tmp = im_m * -re;
	else
		tmp = (im_m * (2.0 + (im_m * (1.0 - (im_m * 0.3333333333333333))))) - 52.0;
	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, 1.55e+68], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(2.0 + N[(im$95$m * N[(1.0 - N[(im$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.5499999999999999e68

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 68.3%

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

      1. associate-*r* 68.3%

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

      2. neg-mul-1 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in re around 0 39.5%

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

      1. associate-*r* 39.5%

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

      2. mul-1-neg 39.5%

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

   8. Simplified 39.5%

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

   if 1.5499999999999999e68 < im

   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 48.9%

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

   6. Applied egg-rr 48.9%

      \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 43.0%

      \[\leadsto \color{blue}{im \cdot \left(2 + im \cdot \left(1 + 0.3333333333333333 \cdot im\right)\right) - 52} \]
   8. Step-by-step derivation

      1. *-commutative 43.0%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \color{blue}{im \cdot 0.3333333333333333}\right)\right) - 52 \]

      2. add-sqr-sqrt 43.0%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot 0.3333333333333333\right)\right) - 52 \]

      3. sqrt-prod 43.0%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \color{blue}{\sqrt{im \cdot im}} \cdot 0.3333333333333333\right)\right) - 52 \]

      4. sqr-neg 43.0%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot 0.3333333333333333\right)\right) - 52 \]

      5. sqrt-unprod 0.0%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot 0.3333333333333333\right)\right) - 52 \]

      6. add-sqr-sqrt 43.2%

         \[\leadsto im \cdot \left(2 + im \cdot \left(1 + \color{blue}{\left(-im\right)} \cdot 0.3333333333333333\right)\right) - 52 \]

      7. cancel-sign-sub-inv 43.2%

         \[\leadsto im \cdot \left(2 + im \cdot \color{blue}{\left(1 - im \cdot 0.3333333333333333\right)}\right) - 52 \]

   9. Applied egg-rr 43.2%

      \[\leadsto im \cdot \left(2 + im \cdot \color{blue}{\left(1 - im \cdot 0.3333333333333333\right)}\right) - 52 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.55 \cdot 10^{+68}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 - im \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 + im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \end{array} \]

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 2.2e+100)
    (* im_m (- re))
    (- (* im_m (+ 2.0 (* im_m (+ 1.0 (* im_m 0.3333333333333333))))) 52.0))))

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 <= 2.2e+100) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	}
	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 <= 2.2d+100) then
        tmp = im_m * -re
    else
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 + (im_m * 0.3333333333333333d0))))) - 52.0d0
    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 <= 2.2e+100) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	}
	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 <= 2.2e+100:
		tmp = im_m * -re
	else:
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0
	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 <= 2.2e+100)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.3333333333333333))))) - 52.0);
	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 <= 2.2e+100)
		tmp = im_m * -re;
	else
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	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, 2.2e+100], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(2.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.2000000000000001e100

   1. Initial program 54.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

   6. Taylor expanded in re around 0 38.5%

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

      1. associate-*r* 38.5%

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

      2. mul-1-neg 38.5%

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

   8. Simplified 38.5%

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

   if 2.2000000000000001e100 < im

   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 48.8%

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

   6. Applied egg-rr 48.8%

      \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 48.8%

      \[\leadsto \color{blue}{im \cdot \left(2 + im \cdot \left(1 + 0.3333333333333333 \cdot im\right)\right) - 52} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 + im \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.0% accurate, 19.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \end{array} \]

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 1.1e+102)
    (* im_m (- re))
    (- (* im_m (+ 2.0 (* im_m (* im_m 0.3333333333333333)))) 52.0))))

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 <= 1.1e+102) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (im_m * 0.3333333333333333)))) - 52.0;
	}
	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 <= 1.1d+102) then
        tmp = im_m * -re
    else
        tmp = (im_m * (2.0d0 + (im_m * (im_m * 0.3333333333333333d0)))) - 52.0d0
    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 <= 1.1e+102) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (2.0 + (im_m * (im_m * 0.3333333333333333)))) - 52.0;
	}
	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 <= 1.1e+102:
		tmp = im_m * -re
	else:
		tmp = (im_m * (2.0 + (im_m * (im_m * 0.3333333333333333)))) - 52.0
	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 <= 1.1e+102)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(im_m * 0.3333333333333333)))) - 52.0);
	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 <= 1.1e+102)
		tmp = im_m * -re;
	else
		tmp = (im_m * (2.0 + (im_m * (im_m * 0.3333333333333333)))) - 52.0;
	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, 1.1e+102], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(2.0 + N[(im$95$m * N[(im$95$m * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.10000000000000004e102

   1. Initial program 54.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

   6. Taylor expanded in re around 0 38.5%

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

      1. associate-*r* 38.5%

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

      2. mul-1-neg 38.5%

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

   8. Simplified 38.5%

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

   if 1.10000000000000004e102 < im

   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 48.8%

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

   6. Applied egg-rr 48.8%

      \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 48.8%

      \[\leadsto \color{blue}{im \cdot \left(2 + im \cdot \left(1 + 0.3333333333333333 \cdot im\right)\right) - 52} \]

   8. Taylor expanded in im around inf 48.8%

      \[\leadsto im \cdot \left(2 + im \cdot \color{blue}{\left(0.3333333333333333 \cdot im\right)}\right) - 52 \]
   9. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto im \cdot \left(2 + im \cdot \color{blue}{\left(im \cdot 0.3333333333333333\right)}\right) - 52 \]

   10. Simplified 48.8%

       \[\leadsto im \cdot \left(2 + im \cdot \color{blue}{\left(im \cdot 0.3333333333333333\right)}\right) - 52 \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(im \cdot 0.3333333333333333\right)\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.3% accurate, 25.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.15 \cdot 10^{+186}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m + 2\right) - 52\\ \end{array} \end{array} \]

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 2.15e+186) (* im_m (- re)) (- (* im_m (+ im_m 2.0)) 52.0))))

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 <= 2.15e+186) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (im_m + 2.0)) - 52.0;
	}
	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 <= 2.15d+186) then
        tmp = im_m * -re
    else
        tmp = (im_m * (im_m + 2.0d0)) - 52.0d0
    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 <= 2.15e+186) {
		tmp = im_m * -re;
	} else {
		tmp = (im_m * (im_m + 2.0)) - 52.0;
	}
	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 <= 2.15e+186:
		tmp = im_m * -re
	else:
		tmp = (im_m * (im_m + 2.0)) - 52.0
	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 <= 2.15e+186)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(Float64(im_m * Float64(im_m + 2.0)) - 52.0);
	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 <= 2.15e+186)
		tmp = im_m * -re;
	else
		tmp = (im_m * (im_m + 2.0)) - 52.0;
	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, 2.15e+186], N[(im$95$m * (-re)), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m + 2.0), $MachinePrecision]), $MachinePrecision] - 52.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.15e186

   1. Initial program 58.0%

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

   3. Taylor expanded in im around 0 61.4%

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

      1. associate-*r* 61.4%

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

      2. neg-mul-1 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in re around 0 36.8%

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

      1. associate-*r* 36.8%

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

      2. mul-1-neg 36.8%

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

   8. Simplified 36.8%

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

   if 2.15e186 < im

   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 45.5%

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

   6. Applied egg-rr 45.5%

      \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 45.5%

      \[\leadsto \color{blue}{im \cdot \left(2 + im\right) - 52} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.15 \cdot 10^{+186}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im + 2\right) - 52\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.7% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8.5 \cdot 10^{+239}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

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 8.5e+239) (* 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 (im_m <= 8.5e+239) {
		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 (im_m <= 8.5d+239) 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 (im_m <= 8.5e+239) {
		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 im_m <= 8.5e+239:
		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 (im_m <= 8.5e+239)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = 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 <= 8.5e+239)
		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[im$95$m, 8.5e+239], N[(im$95$m * (-re)), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 8.50000000000000021e239

   1. Initial program 59.4%

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

   3. Taylor expanded in im around 0 59.5%

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

      1. associate-*r* 59.5%

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

      2. neg-mul-1 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in re around 0 36.1%

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

      1. associate-*r* 36.1%

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

      2. mul-1-neg 36.1%

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

   8. Simplified 36.1%

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

   if 8.50000000000000021e239 < 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 6.1%

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

      1. associate-*r* 6.1%

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

      2. neg-mul-1 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in re around 0 9.5%

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

      1. associate-*r* 9.5%

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

      2. mul-1-neg 9.5%

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

   8. Simplified 9.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 35.7%

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

      3. sqr-neg 35.7%

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

      4. sqrt-prod 36.0%

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

      5. add-sqr-sqrt 36.0%

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

      6. pow1 36.0%

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

   10. Applied egg-rr 36.0%

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

       1. unpow1 36.0%

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

   12. Simplified 36.0%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.5 \cdot 10^{+239}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 20.3% 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 61.6%

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

3. Taylor expanded in im around 0 56.6%

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

   1. associate-*r* 56.6%

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

   2. neg-mul-1 56.6%

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

5. Simplified 56.6%

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

6. Taylor expanded in re around 0 34.6%

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

   1. associate-*r* 34.6%

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

   2. mul-1-neg 34.6%

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

8. Simplified 34.6%

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

   1. add-sqr-sqrt 17.2%

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

   2. sqrt-unprod 32.1%

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

   3. sqr-neg 32.1%

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

   4. sqrt-prod 9.6%

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

   5. add-sqr-sqrt 21.0%

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

   6. pow1 21.0%

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

10. Applied egg-rr 21.0%

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

    1. unpow1 21.0%

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

12. Simplified 21.0%

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

Alternative 13: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -52 \end{array} \]

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 -52.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -52.0;
}

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 * (-52.0d0)
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 * -52.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -52.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -52.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -52.0;
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 * -52.0), $MachinePrecision]

Derivation

1. Initial program 61.6%

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

4. Applied egg-rr 38.7%

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

6. Applied egg-rr 13.3%

   \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

7. Taylor expanded in im around 0 2.9%

   \[\leadsto \color{blue}{-52} \]
8. 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 2024129 
(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))))