math.cos on complex, imaginary part

Percentage Accurate: 64.8% → 99.9%

Time: 8.7s

Alternatives: 13

Speedup: 2.8×

• 48.2% 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: 64.8% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -0.4)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (* im_m im_m)
          (-
           (*
            (* im_m im_m)
            (- (* (* im_m im_m) -0.0003968253968253968) 0.016666666666666666))
           0.3333333333333333))
         2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-0.4d0)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -0.4:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.4)
		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) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -0.4)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.4], 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] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.40000000000000002

   1. Initial program 100.0%

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

   if -0.40000000000000002 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.7%

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

   3. Taylor expanded in im around 0 95.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 95.1%

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

   5. Applied egg-rr 95.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 95.1%

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

   7. Applied egg-rr 95.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 95.1%

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

   9. Applied egg-rr 95.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.4:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 210 \lor \neg \left(im\_m \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot 8\\ \end{array} \end{array} \]

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 210.0) (not (<= im_m 3.35e+44)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.0003968253968253968) 0.016666666666666666))
         0.3333333333333333))
       2.0)))
    (* (- (exp (- im_m)) (exp im_m)) 8.0))))

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 <= 210.0) || !(im_m <= 3.35e+44)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (exp(-im_m) - exp(im_m)) * 8.0;
	}
	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 <= 210.0d0) .or. (.not. (im_m <= 3.35d+44))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    else
        tmp = (exp(-im_m) - exp(im_m)) * 8.0d0
    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 <= 210.0) || !(im_m <= 3.35e+44)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	} else {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * 8.0;
	}
	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 <= 210.0) or not (im_m <= 3.35e+44):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	else:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * 8.0
	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 <= 210.0) || !(im_m <= 3.35e+44))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	else
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * 8.0);
	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 <= 210.0) || ~((im_m <= 3.35e+44)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	else
		tmp = (exp(-im_m) - exp(im_m)) * 8.0;
	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[Or[LessEqual[im$95$m, 210.0], N[Not[LessEqual[im$95$m, 3.35e+44]], $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] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 210 or 3.35000000000000018e44 < im

   1. Initial program 66.2%

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

   3. Taylor expanded in im around 0 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.3%

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

   5. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 96.3%

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

   7. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 96.3%

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

   9. Applied egg-rr 96.3%

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

   if 210 < im < 3.35000000000000018e44

   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 41.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 210 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot 8\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 140 \lor \neg \left(im\_m \leq 4 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 140.0) (not (<= im_m 4e+44)))
    (*
     (* 0.5 (sin re))
     (*
      im_m
      (-
       (*
        (* im_m im_m)
        (-
         (*
          (* im_m im_m)
          (- (* (* im_m im_m) -0.0003968253968253968) 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 <= 140.0) || !(im_m <= 4e+44)) {
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 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 <= 140.0d0) .or. (.not. (im_m <= 4d+44))) then
        tmp = (0.5d0 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 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 <= 140.0) || !(im_m <= 4e+44)) {
		tmp = (0.5 * Math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 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 <= 140.0) or not (im_m <= 4e+44):
		tmp = (0.5 * math.sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 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 <= 140.0) || !(im_m <= 4e+44))
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 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 <= 140.0) || ~((im_m <= 4e+44)))
		tmp = (0.5 * sin(re)) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 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, 140.0], N[Not[LessEqual[im$95$m, 4e+44]], $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] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $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 < 140 or 4.0000000000000004e44 < im

   1. Initial program 66.2%

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

   3. Taylor expanded in im around 0 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.3%

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

   5. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 96.3%

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

   7. Applied egg-rr 96.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 96.3%

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

   9. Applied egg-rr 96.3%

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

   if 140 < im < 4.0000000000000004e44

   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 41.2%

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

   6. Applied egg-rr 41.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 140 \lor \neg \left(im \leq 4 \cdot 10^{+44}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

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 3.1e+37)
    (* im_m (- (sin re)))
    (if (<= im_m 6.2e+130)
      (* (- 27.0 (exp im_m)) -2.0)
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

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 <= 3.1e+37) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 6.2e+130) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 3.1d+37) then
        tmp = im_m * -sin(re)
    else if (im_m <= 6.2d+130) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 3.1e+37) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 6.2e+130) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 3.1e+37:
		tmp = im_m * -math.sin(re)
	elif im_m <= 6.2e+130:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 3.1e+37)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 6.2e+130)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 3.1e+37)
		tmp = im_m * -sin(re);
	elseif (im_m <= 6.2e+130)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 3.1e+37], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 6.2e+130], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.1000000000000002e37

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0 63.2%

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

      1. associate-*r* 63.2%

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

      2. neg-mul-1 63.2%

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

   5. Simplified 63.2%

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

   if 3.1000000000000002e37 < im < 6.1999999999999999e130

   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 30.8%

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

   6. Applied egg-rr 30.8%

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

   if 6.1999999999999999e130 < 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 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in im around 0 36.0%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 75:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (<= im_m 75.0) (* 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 <= 75.0) {
		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 <= 75.0d0) 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 <= 75.0) {
		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 <= 75.0:
		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 <= 75.0)
		tmp = Float64(im_m * Float64(-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 <= 75.0)
		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, 75.0], 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 < 75

   1. Initial program 54.9%

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

   3. Taylor expanded in im around 0 67.9%

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

      1. associate-*r* 67.9%

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

      2. neg-mul-1 67.9%

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

   5. Simplified 67.9%

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

   if 75 < 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 42.9%

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

   6. Applied egg-rr 42.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 75:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 14200000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 + im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

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 14200000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 2.6e+99)
      (* im_m (- re))
      (if (<= im_m 6.2e+130)
        (- (* im_m (+ 2.0 (* im_m (+ 1.0 (* im_m 0.3333333333333333))))) 52.0)
        (+
         208.0
         (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0))))))))

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 <= 14200000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 2.6e+99) {
		tmp = im_m * -re;
	} else if (im_m <= 6.2e+130) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 14200000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 2.6d+99) then
        tmp = im_m * -re
    else if (im_m <= 6.2d+130) then
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 + (im_m * 0.3333333333333333d0))))) - 52.0d0
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 14200000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 2.6e+99) {
		tmp = im_m * -re;
	} else if (im_m <= 6.2e+130) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 14200000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 2.6e+99:
		tmp = im_m * -re
	elif im_m <= 6.2e+130:
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 14200000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 2.6e+99)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 6.2e+130)
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.3333333333333333))))) - 52.0);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 14200000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 2.6e+99)
		tmp = im_m * -re;
	elseif (im_m <= 6.2e+130)
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 14200000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.6e+99], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 6.2e+130], 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], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 1.42e10

   1. Initial program 55.7%

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

   3. Taylor expanded in im around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 1.42e10 < im < 2.6e99

   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 2.9%

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

      1. associate-*r* 2.9%

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

      2. neg-mul-1 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in re around 0 11.1%

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

      1. associate-*r* 11.1%

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

      2. mul-1-neg 11.1%

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

   8. Simplified 11.1%

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

   if 2.6e99 < im < 6.1999999999999999e130

   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 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if 6.1999999999999999e130 < 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 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in im around 0 36.0%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 14200000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 + im \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.4 \cdot 10^{+99}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;im\_m \cdot \left(2 + im\_m \cdot \left(1 + im\_m \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

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 2.4e+99)
    (* im_m (- re))
    (if (<= im_m 6.2e+130)
      (- (* im_m (+ 2.0 (* im_m (+ 1.0 (* im_m 0.3333333333333333))))) 52.0)
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

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 <= 2.4e+99) {
		tmp = im_m * -re;
	} else if (im_m <= 6.2e+130) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 2.4d+99) then
        tmp = im_m * -re
    else if (im_m <= 6.2d+130) then
        tmp = (im_m * (2.0d0 + (im_m * (1.0d0 + (im_m * 0.3333333333333333d0))))) - 52.0d0
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 2.4e+99) {
		tmp = im_m * -re;
	} else if (im_m <= 6.2e+130) {
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 2.4e+99:
		tmp = im_m * -re
	elif im_m <= 6.2e+130:
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 2.4e+99)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 6.2e+130)
		tmp = Float64(Float64(im_m * Float64(2.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.3333333333333333))))) - 52.0);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 2.4e+99)
		tmp = im_m * -re;
	elseif (im_m <= 6.2e+130)
		tmp = (im_m * (2.0 + (im_m * (1.0 + (im_m * 0.3333333333333333))))) - 52.0;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 2.4e+99], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 6.2e+130], 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], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.4000000000000001e99

   1. Initial program 60.6%

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

   3. Taylor expanded in im around 0 59.7%

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

      1. associate-*r* 59.7%

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

      2. neg-mul-1 59.7%

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

   5. Simplified 59.7%

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

   6. Taylor expanded in re around 0 37.0%

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

      1. associate-*r* 37.0%

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

      2. mul-1-neg 37.0%

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

   8. Simplified 37.0%

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

   if 2.4000000000000001e99 < im < 6.1999999999999999e130

   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 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if 6.1999999999999999e130 < 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 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in im around 0 36.0%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4 \cdot 10^{+99}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 6.2 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(2 + im \cdot \left(1 + im \cdot 0.3333333333333333\right)\right) - 52\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

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 1.6e+118)
    (* im_m (- re))
    (+ 208.0 (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0))))))

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 <= 1.6e+118) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 1.6d+118) then
        tmp = im_m * -re
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 1.6e+118) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 1.6e+118:
		tmp = im_m * -re
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 1.6e+118)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 1.6e+118)
		tmp = im_m * -re;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 1.6e+118], N[(im$95$m * (-re)), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.60000000000000008e118

   1. Initial program 60.8%

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

   3. Taylor expanded in im around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. neg-mul-1 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in re around 0 36.9%

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

      1. associate-*r* 36.9%

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

      2. mul-1-neg 36.9%

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

   8. Simplified 36.9%

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

   if 1.60000000000000008e118 < 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 36.0%

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

   6. Applied egg-rr 36.0%

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

   7. Taylor expanded in im around 0 36.0%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.6 \cdot 10^{+118}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.4% accurate, 25.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.9 \cdot 10^{+150}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot -4\right)\\ \end{array} \end{array} \]

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 2.9e+150) (* im_m (- re)) (+ 208.0 (* im_m (* im_m -4.0))))))

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 <= 2.9e+150) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * (im_m * -4.0));
	}
	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 <= 2.9d+150) then
        tmp = im_m * -re
    else
        tmp = 208.0d0 + (im_m * (im_m * (-4.0d0)))
    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 <= 2.9e+150) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * (im_m * -4.0));
	}
	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 <= 2.9e+150:
		tmp = im_m * -re
	else:
		tmp = 208.0 + (im_m * (im_m * -4.0))
	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 <= 2.9e+150)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(im_m * -4.0)));
	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 <= 2.9e+150)
		tmp = im_m * -re;
	else
		tmp = 208.0 + (im_m * (im_m * -4.0));
	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, 2.9e+150], N[(im$95$m * (-re)), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(im$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.90000000000000011e150

   1. Initial program 61.0%

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

   3. Taylor expanded in im around 0 59.1%

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

      1. associate-*r* 59.1%

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

      2. neg-mul-1 59.1%

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

   5. Simplified 59.1%

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

   6. Taylor expanded in re around 0 36.7%

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

      1. associate-*r* 36.7%

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

      2. mul-1-neg 36.7%

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

   8. Simplified 36.7%

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

   if 2.90000000000000011e150 < 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 34.7%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in im around 0 34.7%

      \[\leadsto \color{blue}{208 + im \cdot \left(-4 \cdot im - 8\right)} \]

   8. Taylor expanded in im around inf 34.7%

      \[\leadsto 208 + im \cdot \color{blue}{\left(-4 \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 34.7%

         \[\leadsto 208 + im \cdot \color{blue}{\left(im \cdot -4\right)} \]

   10. Simplified 34.7%

       \[\leadsto 208 + im \cdot \color{blue}{\left(im \cdot -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.4% 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 #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 * 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 68.5%

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

3. Taylor expanded in im around 0 48.9%

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

   1. associate-*r* 48.9%

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

   2. neg-mul-1 48.9%

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

5. Simplified 48.9%

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

6. Taylor expanded in re around 0 33.8%

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

   1. associate-*r* 33.8%

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

   2. mul-1-neg 33.8%

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

8. Simplified 33.8%

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

9. Final simplification 33.8%

   \[\leadsto im \cdot \left(-re\right) \]
10. Add Preprocessing

Alternative 11: 20.9% 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 68.5%

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

3. Taylor expanded in im around 0 48.9%

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

   1. associate-*r* 48.9%

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

   2. neg-mul-1 48.9%

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

5. Simplified 48.9%

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

6. Taylor expanded in re around 0 33.8%

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

   1. associate-*r* 33.8%

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

   2. mul-1-neg 33.8%

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

8. Simplified 33.8%

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

   1. add-sqr-sqrt 17.6%

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

   2. sqrt-unprod 30.9%

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

   3. sqr-neg 30.9%

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

   4. pow2 30.9%

      \[\leadsto \sqrt{\color{blue}{{im}^{2}}} \cdot re \]

   5. sqrt-pow1 21.3%

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

   6. metadata-eval 21.3%

      \[\leadsto {im}^{\color{blue}{1}} \cdot re \]

   7. pow1 21.3%

      \[\leadsto {im}^{1} \cdot \color{blue}{{re}^{1}} \]

   8. pow-prod-down 21.3%

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

10. Applied egg-rr 21.3%

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

    1. unpow1 21.3%

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

12. Simplified 21.3%

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

Alternative 12: 5.4% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -16\right) \end{array} \]

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 -16.0)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -16.0);
}

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 * (im_m * (-16.0d0))
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 * (im_m * -16.0);
}

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 * -16.0)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * -16.0))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -16.0);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * -16.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

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

4. Applied egg-rr 38.5%

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

5. Taylor expanded in im around 0 4.8%

   \[\leadsto 8 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

6. Taylor expanded in im around 0 4.8%

   \[\leadsto \color{blue}{-16 \cdot im} \]
7. Step-by-step derivation

   1. *-commutative 4.8%

      \[\leadsto \color{blue}{im \cdot -16} \]

8. Simplified 4.8%

   \[\leadsto \color{blue}{im \cdot -16} \]
9. 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 68.5%

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

4. Applied egg-rr 43.6%

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

6. Applied egg-rr 19.2%

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

7. Taylor expanded in im around 0 2.5%

   \[\leadsto \color{blue}{-52} \]
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 2024172 
(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))))