math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.7%

Time: 10.6s

Alternatives: 17

Speedup: 2.5×

• 49.5% 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: 65.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.5d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.5

   1. Initial program 100.0%

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

   if -0.5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. *-commutative 87.8%

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

      5. associate-*r* 87.8%

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

      6. distribute-lft-out-- 87.8%

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

      7. associate-*r* 87.8%

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

      8. *-commutative 87.8%

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

      9. associate-*r* 87.8%

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

      10. associate-*r* 89.7%

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

      11. distribute-rgt-out-- 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 92.4%

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

Alternative 2: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 450000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 9.8 \cdot 10^{+24} \lor \neg \left(im\_m \leq 6 \cdot 10^{+249}\right) \land im\_m \leq 6.8 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 450000000.0)
    (* im_m (- (sin re)))
    (if (or (<= im_m 9.8e+24) (and (not (<= im_m 6e+249)) (<= im_m 6.8e+259)))
      (log1p (expm1 (* im_m 0.16666666666666666)))
      (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 450000000.0) {
		tmp = im_m * -sin(re);
	} else if ((im_m <= 9.8e+24) || (!(im_m <= 6e+249) && (im_m <= 6.8e+259))) {
		tmp = log1p(expm1((im_m * 0.16666666666666666)));
	} else {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 450000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if ((im_m <= 9.8e+24) || (!(im_m <= 6e+249) && (im_m <= 6.8e+259))) {
		tmp = Math.log1p(Math.expm1((im_m * 0.16666666666666666)));
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 450000000.0:
		tmp = im_m * -math.sin(re)
	elif (im_m <= 9.8e+24) or (not (im_m <= 6e+249) and (im_m <= 6.8e+259)):
		tmp = math.log1p(math.expm1((im_m * 0.16666666666666666)))
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 450000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif ((im_m <= 9.8e+24) || (!(im_m <= 6e+249) && (im_m <= 6.8e+259)))
		tmp = log1p(expm1(Float64(im_m * 0.16666666666666666)));
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 450000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 9.8e+24], And[N[Not[LessEqual[im$95$m, 6e+249]], $MachinePrecision], LessEqual[im$95$m, 6.8e+259]]], N[Log[1 + N[(Exp[N[(im$95$m * 0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e8

   1. Initial program 59.6%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 4.5e8 < im < 9.80000000000000059e24 or 6.00000000000000032e249 < im < 6.79999999999999979e259

   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 61.4%

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

      1. associate-*r* 61.4%

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

      2. distribute-rgt-out 61.4%

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

      3. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   7. Applied egg-rr 3.6%

      \[\leadsto im \cdot \color{blue}{0.16666666666666666} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 60.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)} \]

   9. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)} \]

   if 9.80000000000000059e24 < im < 6.00000000000000032e249 or 6.79999999999999979e259 < 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 66.4%

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

      1. +-commutative 66.4%

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

      2. mul-1-neg 66.4%

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

      3. unsub-neg 66.4%

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

      4. *-commutative 66.4%

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

      5. associate-*r* 66.4%

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

      6. distribute-lft-out-- 66.4%

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

      7. associate-*r* 66.4%

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

      8. *-commutative 66.4%

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

      9. associate-*r* 66.4%

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

      10. associate-*r* 74.3%

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

      11. distribute-rgt-out-- 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in re around 0 64.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+24} \lor \neg \left(im \leq 6 \cdot 10^{+249}\right) \land im \leq 6.8 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 580:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\_m\right) \cdot {\sin re}^{-3}\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+249} \lor \neg \left(im\_m \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 580.0)
    (* im_m (- (sin re)))
    (if (<= im_m 5.6e+102)
      (* (- im_m) (pow (sin re) -3.0))
      (if (or (<= im_m 6e+249) (not (<= im_m 6.8e+259)))
        (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
        (log1p (expm1 (* im_m 0.16666666666666666))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 580.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else if ((im_m <= 6e+249) || !(im_m <= 6.8e+259)) {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = log1p(expm1((im_m * 0.16666666666666666)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 580.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else if ((im_m <= 6e+249) || !(im_m <= 6.8e+259)) {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = Math.log1p(Math.expm1((im_m * 0.16666666666666666)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 580.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 5.6e+102:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	elif (im_m <= 6e+249) or not (im_m <= 6.8e+259):
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	else:
		tmp = math.log1p(math.expm1((im_m * 0.16666666666666666)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 580.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	elseif ((im_m <= 6e+249) || !(im_m <= 6.8e+259))
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	else
		tmp = log1p(expm1(Float64(im_m * 0.16666666666666666)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 580.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 6e+249], N[Not[LessEqual[im$95$m, 6.8e+259]], $MachinePrecision]], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(im$95$m * 0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 580

   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 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   5. Simplified 64.8%

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

   if 580 < im < 5.60000000000000037e102

   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 3.2%

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

      1. associate-*r* 3.2%

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

      2. neg-mul-1 3.2%

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

   5. Simplified 3.2%

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

   7. Applied egg-rr 39.2%

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

   if 5.60000000000000037e102 < im < 6.00000000000000032e249 or 6.79999999999999979e259 < 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 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. *-commutative 89.2%

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

      5. associate-*r* 89.2%

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

      6. distribute-lft-out-- 89.2%

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

      7. associate-*r* 89.2%

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

      8. *-commutative 89.2%

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

      9. associate-*r* 89.2%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out-- 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 74.4%

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

   if 6.00000000000000032e249 < im < 6.79999999999999979e259

   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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 4.8%

      \[\leadsto im \cdot \color{blue}{0.16666666666666666} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 66.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)} \]

   9. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+249} \lor \neg \left(im \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := {im\_m}^{3} \cdot -0.16666666666666666\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.2:\\ \;\;\;\;\sin re \cdot \left(t\_0 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (pow im_m 3.0) -0.16666666666666666)))
   (*
    im_s
    (if (<= im_m 0.2)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 5.6e+102)
        (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
        (* (sin re) t_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 = pow(im_m, 3.0) * -0.16666666666666666;
	double tmp;
	if (im_m <= 0.2) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * t_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) :: tmp
    t_0 = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    if (im_m <= 0.2d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 5.6d+102) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * t_0
    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.pow(im_m, 3.0) * -0.16666666666666666;
	double tmp;
	if (im_m <= 0.2) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * t_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.pow(im_m, 3.0) * -0.16666666666666666
	tmp = 0
	if im_m <= 0.2:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 5.6e+102:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * t_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((im_m ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im_m <= 0.2)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * t_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 = (im_m ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im_m <= 0.2)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 5.6e+102)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * t_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[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 0.2], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 0.20000000000000001

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. *-commutative 87.8%

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

      5. associate-*r* 87.8%

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

      6. distribute-lft-out-- 87.8%

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

      7. associate-*r* 87.8%

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

      8. *-commutative 87.8%

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

      9. associate-*r* 87.8%

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

      10. associate-*r* 89.7%

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

      11. distribute-rgt-out-- 89.7%

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

   5. Simplified 89.7%

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

   if 0.20000000000000001 < im < 5.60000000000000037e102

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

   5. Simplified 77.1%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.9%

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

      1. associate-*r* 89.9%

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

      2. distribute-rgt-out 89.9%

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

      3. *-commutative 89.9%

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

   5. Simplified 89.9%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = sin(re) * (pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 440.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 5.6d+102) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = sin(re) * ((im_m ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.sin(re) * (Math.pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 440.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 5.6e+102:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.sin(re) * (math.pow(im_m, 3.0) * -0.16666666666666666)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 440.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(sin(re) * Float64((im_m ^ 3.0) * -0.16666666666666666));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 440.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 5.6e+102)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = sin(re) * ((im_m ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 440.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 440

   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 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   5. Simplified 64.8%

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

   if 440 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.2%

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

      1. associate-*r* 3.2%

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

      2. neg-mul-1 3.2%

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

   5. Simplified 3.2%

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

   7. Applied egg-rr 39.2%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.9%

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

      1. associate-*r* 89.9%

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

      2. distribute-rgt-out 89.9%

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

      3. *-commutative 89.9%

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

   5. Simplified 89.9%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.2%

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

Alternative 6: 89.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := {im\_m}^{3} \cdot -0.16666666666666666\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 420:\\ \;\;\;\;\sin re \cdot \left(t\_0 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\_m\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (pow im_m 3.0) -0.16666666666666666)))
   (*
    im_s
    (if (<= im_m 420.0)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 5.6e+102)
        (* (- im_m) (pow (sin re) -3.0))
        (* (sin re) t_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 = pow(im_m, 3.0) * -0.16666666666666666;
	double tmp;
	if (im_m <= 420.0) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * pow(sin(re), -3.0);
	} else {
		tmp = sin(re) * t_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) :: tmp
    t_0 = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    if (im_m <= 420.0d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 5.6d+102) then
        tmp = -im_m * (sin(re) ** (-3.0d0))
    else
        tmp = sin(re) * t_0
    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.pow(im_m, 3.0) * -0.16666666666666666;
	double tmp;
	if (im_m <= 420.0) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = -im_m * Math.pow(Math.sin(re), -3.0);
	} else {
		tmp = Math.sin(re) * t_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.pow(im_m, 3.0) * -0.16666666666666666
	tmp = 0
	if im_m <= 420.0:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 5.6e+102:
		tmp = -im_m * math.pow(math.sin(re), -3.0)
	else:
		tmp = math.sin(re) * t_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((im_m ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im_m <= 420.0)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(Float64(-im_m) * (sin(re) ^ -3.0));
	else
		tmp = Float64(sin(re) * t_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 = (im_m ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im_m <= 420.0)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 5.6e+102)
		tmp = -im_m * (sin(re) ^ -3.0);
	else
		tmp = sin(re) * t_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[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 420.0], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[((-im$95$m) * N[Power[N[Sin[re], $MachinePrecision], -3.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 420

   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 86.9%

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

      1. +-commutative 86.9%

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

      2. mul-1-neg 86.9%

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

      3. unsub-neg 86.9%

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

      4. *-commutative 86.9%

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

      5. associate-*r* 86.9%

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

      6. distribute-lft-out-- 86.9%

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

      7. associate-*r* 86.9%

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

      8. *-commutative 86.9%

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

      9. associate-*r* 86.9%

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

      10. associate-*r* 88.7%

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

      11. distribute-rgt-out-- 88.7%

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

   5. Simplified 88.7%

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

   if 420 < im < 5.60000000000000037e102

   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 3.2%

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

      1. associate-*r* 3.2%

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

      2. neg-mul-1 3.2%

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

   5. Simplified 3.2%

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

   7. Applied egg-rr 39.2%

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

   if 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 89.9%

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

      1. associate-*r* 89.9%

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

      2. distribute-rgt-out 89.9%

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

      3. *-commutative 89.9%

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

   5. Simplified 89.9%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(-im\right) \cdot {\sin re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00055:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.8 \cdot 10^{+243} \lor \neg \left(im\_m \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(0.16666666666666666 \cdot {re}^{2} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00055)
    (* im_m (- (sin re)))
    (if (or (<= im_m 1.8e+243) (not (<= im_m 6.8e+259)))
      (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
      (* re (* im_m (+ (* 0.16666666666666666 (pow re 2.0)) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00055) {
		tmp = im_m * -sin(re);
	} else if ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259)) {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = re * (im_m * ((0.16666666666666666 * pow(re, 2.0)) + -1.0));
	}
	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 <= 0.00055d0) then
        tmp = im_m * -sin(re)
    else if ((im_m <= 1.8d+243) .or. (.not. (im_m <= 6.8d+259))) then
        tmp = re * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else
        tmp = re * (im_m * ((0.16666666666666666d0 * (re ** 2.0d0)) + (-1.0d0)))
    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 <= 0.00055) {
		tmp = im_m * -Math.sin(re);
	} else if ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259)) {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = re * (im_m * ((0.16666666666666666 * Math.pow(re, 2.0)) + -1.0));
	}
	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 <= 0.00055:
		tmp = im_m * -math.sin(re)
	elif (im_m <= 1.8e+243) or not (im_m <= 6.8e+259):
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	else:
		tmp = re * (im_m * ((0.16666666666666666 * math.pow(re, 2.0)) + -1.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.00055)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259))
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	else
		tmp = Float64(re * Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 2.0)) + -1.0)));
	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 <= 0.00055)
		tmp = im_m * -sin(re);
	elseif ((im_m <= 1.8e+243) || ~((im_m <= 6.8e+259)))
		tmp = re * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	else
		tmp = re * (im_m * ((0.16666666666666666 * (re ^ 2.0)) + -1.0));
	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, 0.00055], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 1.8e+243], N[Not[LessEqual[im$95$m, 6.8e+259]], $MachinePrecision]], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.50000000000000033e-4

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0 65.5%

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

      1. associate-*r* 65.5%

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

      2. neg-mul-1 65.5%

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

   5. Simplified 65.5%

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

   if 5.50000000000000033e-4 < im < 1.7999999999999998e243 or 6.79999999999999979e259 < 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 61.8%

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

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unsub-neg 61.8%

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

      4. *-commutative 61.8%

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

      5. associate-*r* 61.8%

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

      6. distribute-lft-out-- 61.8%

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

      7. associate-*r* 61.8%

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

      8. *-commutative 61.8%

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

      9. associate-*r* 61.8%

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

      10. associate-*r* 68.9%

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

      11. distribute-rgt-out-- 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in re around 0 60.2%

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

   if 1.7999999999999998e243 < im < 6.79999999999999979e259

   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 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in re around 0 69.8%

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

      1. neg-mul-1 69.8%

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

      2. +-commutative 69.8%

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

      3. *-commutative 69.8%

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

      4. associate-*l* 69.8%

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

      5. neg-mul-1 69.8%

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

      6. *-commutative 69.8%

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

      7. distribute-lft-out 69.8%

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

   8. Simplified 69.8%

      \[\leadsto \color{blue}{re \cdot \left(im \cdot \left({re}^{2} \cdot 0.16666666666666666 + -1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00055:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+243} \lor \neg \left(im \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(0.16666666666666666 \cdot {re}^{2} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.3:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.8 \cdot 10^{+243} \lor \neg \left(im\_m \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.3)
    (* im_m (- (sin re)))
    (if (or (<= im_m 1.8e+243) (not (<= im_m 6.8e+259)))
      (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
      (* 0.16666666666666666 (* im_m (pow re 3.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.3) {
		tmp = im_m * -sin(re);
	} else if ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259)) {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.3d0) then
        tmp = im_m * -sin(re)
    else if ((im_m <= 1.8d+243) .or. (.not. (im_m <= 6.8d+259))) then
        tmp = re * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.3) {
		tmp = im_m * -Math.sin(re);
	} else if ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259)) {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.3:
		tmp = im_m * -math.sin(re)
	elif (im_m <= 1.8e+243) or not (im_m <= 6.8e+259):
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	else:
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.3)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif ((im_m <= 1.8e+243) || !(im_m <= 6.8e+259))
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	else
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.3)
		tmp = im_m * -sin(re);
	elseif ((im_m <= 1.8e+243) || ~((im_m <= 6.8e+259)))
		tmp = re * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	else
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.3], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 1.8e+243], N[Not[LessEqual[im$95$m, 6.8e+259]], $MachinePrecision]], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.299999999999999989

   1. Initial program 58.7%

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

   3. Taylor expanded in im around 0 65.5%

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

      1. associate-*r* 65.5%

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

      2. neg-mul-1 65.5%

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

   5. Simplified 65.5%

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

   if 0.299999999999999989 < im < 1.7999999999999998e243 or 6.79999999999999979e259 < 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 61.8%

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

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unsub-neg 61.8%

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

      4. *-commutative 61.8%

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

      5. associate-*r* 61.8%

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

      6. distribute-lft-out-- 61.8%

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

      7. associate-*r* 61.8%

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

      8. *-commutative 61.8%

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

      9. associate-*r* 61.8%

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

      10. associate-*r* 68.9%

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

      11. distribute-rgt-out-- 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in re around 0 60.2%

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

   if 1.7999999999999998e243 < im < 6.79999999999999979e259

   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 7.2%

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

      1. associate-*r* 7.2%

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

      2. neg-mul-1 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in re around 0 69.8%

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

      1. *-commutative 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in re around inf 69.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+243} \lor \neg \left(im \leq 6.8 \cdot 10^{+259}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 560000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 560000000.0)
    (* im_m (- (sin re)))
    (* 0.16666666666666666 (* im_m (pow re 3.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 560000000.0) {
		tmp = im_m * -sin(re);
	} else {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 560000000.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 560000000.0) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 560000000.0:
		tmp = im_m * -math.sin(re)
	else:
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 560000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 560000000.0)
		tmp = im_m * -sin(re);
	else
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 560000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.6e8

   1. Initial program 59.6%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 5.6e8 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.3%

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

      1. associate-*r* 4.3%

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

      2. neg-mul-1 4.3%

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

   5. Simplified 4.3%

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

   6. Taylor expanded in re around 0 25.4%

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

      1. *-commutative 25.4%

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

   8. Simplified 25.4%

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

   9. Taylor expanded in re around inf 24.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 560000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 4.2e+44) (* im_m (- (sin re))) (* (- im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2e+44) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.2d+44) then
        tmp = im_m * -sin(re)
    else
        tmp = -im_m * re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.2e+44) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -im_m * re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.2e+44:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -im_m * re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.2e+44)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(-im_m) * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.2e+44)
		tmp = im_m * -sin(re);
	else
		tmp = -im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.2e+44], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.19999999999999974e44

   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 62.6%

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

      1. associate-*r* 62.6%

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

      2. neg-mul-1 62.6%

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

   5. Simplified 62.6%

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

   if 4.19999999999999974e44 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.5%

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

      1. associate-*r* 4.5%

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

      2. neg-mul-1 4.5%

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

   5. Simplified 4.5%

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

   6. Taylor expanded in re around 0 17.8%

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

      1. associate-*r* 17.8%

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

      2. neg-mul-1 17.8%

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

   8. Simplified 17.8%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.9% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{-25}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 5.2e-25) (* im_m 0.0) (- im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 5.2e-25) {
		tmp = im_m * 0.0;
	} else {
		tmp = -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 (re <= 5.2d-25) then
        tmp = im_m * 0.0d0
    else
        tmp = -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 (re <= 5.2e-25) {
		tmp = im_m * 0.0;
	} else {
		tmp = -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 re <= 5.2e-25:
		tmp = im_m * 0.0
	else:
		tmp = -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 (re <= 5.2e-25)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(-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 (re <= 5.2e-25)
		tmp = im_m * 0.0;
	else
		tmp = -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[re, 5.2e-25], N[(im$95$m * 0.0), $MachinePrecision], (-im$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 5.2e-25

   1. Initial program 73.3%

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

   3. Taylor expanded in im around 0 79.0%

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

      1. associate-*r* 79.0%

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

      2. distribute-rgt-out 79.0%

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

      3. *-commutative 79.0%

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

   5. Simplified 79.0%

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

   7. Applied egg-rr 21.1%

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

   if 5.2e-25 < re

   1. Initial program 59.2%

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

   3. Taylor expanded in im around 0 88.2%

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

      1. associate-*r* 88.2%

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

      2. distribute-rgt-out 88.2%

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

      3. *-commutative 88.2%

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

   5. Simplified 88.2%

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

   7. Applied egg-rr 7.0%

      \[\leadsto im \cdot \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.2 \cdot 10^{-25}:\\ \;\;\;\;im \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 15.5% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 1250000.0) (* im_m 0.0) (* im_m 0.3333333333333333))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.3333333333333333;
	}
	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 (re <= 1250000.0d0) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.3333333333333333d0
    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 (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.3333333333333333;
	}
	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 re <= 1250000.0:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.3333333333333333
	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 (re <= 1250000.0)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.3333333333333333);
	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 (re <= 1250000.0)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.3333333333333333;
	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[re, 1250000.0], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.3333333333333333), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.25e6

   1. Initial program 73.7%

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

   3. Taylor expanded in im around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. distribute-rgt-out 78.8%

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

      3. *-commutative 78.8%

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

   5. Simplified 78.8%

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

   7. Applied egg-rr 20.8%

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

   if 1.25e6 < re

   1. Initial program 57.2%

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

   3. Taylor expanded in im around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. distribute-rgt-out 89.1%

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

      3. *-commutative 89.1%

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

   5. Simplified 89.1%

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

   7. Applied egg-rr 7.4%

      \[\leadsto im \cdot \color{blue}{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 15.5% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 1250000.0) (* im_m 0.0) (* im_m 0.5))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.5;
	}
	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 (re <= 1250000.0d0) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.5d0
    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 (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.5;
	}
	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 re <= 1250000.0:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.5
	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 (re <= 1250000.0)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.5);
	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 (re <= 1250000.0)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.5;
	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[re, 1250000.0], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.25e6

   1. Initial program 73.7%

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

   3. Taylor expanded in im around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. distribute-rgt-out 78.8%

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

      3. *-commutative 78.8%

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

   5. Simplified 78.8%

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

   7. Applied egg-rr 20.8%

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

   if 1.25e6 < re

   1. Initial program 57.2%

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

   3. Taylor expanded in im around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. distribute-rgt-out 89.1%

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

      3. *-commutative 89.1%

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

   5. Simplified 89.1%

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

   7. Applied egg-rr 7.5%

      \[\leadsto im \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 15.6% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.75\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 1250000.0) (* im_m 0.0) (* im_m 0.75))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.75;
	}
	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 (re <= 1250000.0d0) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.75d0
    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 (re <= 1250000.0) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.75;
	}
	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 re <= 1250000.0:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.75
	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 (re <= 1250000.0)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.75);
	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 (re <= 1250000.0)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.75;
	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[re, 1250000.0], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.75), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.25e6

   1. Initial program 73.7%

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

   3. Taylor expanded in im around 0 78.8%

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

      1. associate-*r* 78.8%

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

      2. distribute-rgt-out 78.8%

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

      3. *-commutative 78.8%

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

   5. Simplified 78.8%

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

   7. Applied egg-rr 20.8%

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

   if 1.25e6 < re

   1. Initial program 57.2%

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

   3. Taylor expanded in im around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. distribute-rgt-out 89.1%

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

      3. *-commutative 89.1%

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

   5. Simplified 89.1%

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

   7. Applied egg-rr 8.1%

      \[\leadsto im \cdot \color{blue}{0.75} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1250000:\\ \;\;\;\;im \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot 0.75\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.7% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(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 69.7%

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

3. Taylor expanded in im around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. neg-mul-1 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in re around 0 32.1%

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

   1. associate-*r* 32.1%

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

   2. neg-mul-1 32.1%

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

8. Simplified 32.1%

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

9. Final simplification 32.1%

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

Alternative 16: 5.6% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -3.0);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in im around 0 81.3%

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

   1. associate-*r* 81.3%

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

   2. distribute-rgt-out 81.3%

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

   3. *-commutative 81.3%

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

5. Simplified 81.3%

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

7. Applied egg-rr 5.1%

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

8. Final simplification 5.1%

   \[\leadsto im \cdot -3 \]
9. Add Preprocessing

Alternative 17: 6.2% accurate, 154.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

3. Taylor expanded in im around 0 81.3%

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

   1. associate-*r* 81.3%

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

   2. distribute-rgt-out 81.3%

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

   3. *-commutative 81.3%

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

5. Simplified 81.3%

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

7. Applied egg-rr 5.6%

   \[\leadsto im \cdot \color{blue}{-1} \]

8. Final simplification 5.6%

   \[\leadsto -im \]
9. Add Preprocessing

Developer target: 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 2024055 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

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

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