math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.1%

Time: 18.9s

Alternatives: 27

Speedup: 2.2×

• 49.0% 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: 65.1% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.6e-15)
    (* (- im_m) (sin re))
    (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.6d-15) then
        tmp = -im_m * sin(re)
    else
        tmp = (sin(re) * 0.5d0) * (exp(-im_m) - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (Math.sin(re) * 0.5) * (Math.exp(-im_m) - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (math.sin(re) * 0.5) * (math.exp(-im_m) - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.6e-15)
		tmp = -im_m * sin(re);
	else
		tmp = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im

   1. Initial program 98.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{{im\_m}^{2} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot -0.004629629629629629\right)\right)\\ \mathbf{elif}\;im\_m \leq 6.5 \cdot 10^{+252} \lor \neg \left(im\_m \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left(-im\_m\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.6e-15)
    (* (- im_m) (sin re))
    (if (<= im_m 2.7e+208)
      (log1p (expm1 (* im_m -0.16666666666666666)))
      (if (<= im_m 1.58e+227)
        (sqrt (* (pow im_m 2.0) 0.027777777777777776))
        (if (<= im_m 7.2e+239)
          (log1p (expm1 (* im_m -0.004629629629629629)))
          (if (or (<= im_m 6.5e+252) (not (<= im_m 6.6e+273)))
            (* (- im_m) (+ re (* -0.16666666666666666 (pow re 3.0))))
            (* -0.16666666666666666 (* re (pow im_m 3.0))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.7e+208) {
		tmp = log1p(expm1((im_m * -0.16666666666666666)));
	} else if (im_m <= 1.58e+227) {
		tmp = sqrt((pow(im_m, 2.0) * 0.027777777777777776));
	} else if (im_m <= 7.2e+239) {
		tmp = log1p(expm1((im_m * -0.004629629629629629)));
	} else if ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = -im_m * (re + (-0.16666666666666666 * pow(re, 3.0)));
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.7e+208) {
		tmp = Math.log1p(Math.expm1((im_m * -0.16666666666666666)));
	} else if (im_m <= 1.58e+227) {
		tmp = Math.sqrt((Math.pow(im_m, 2.0) * 0.027777777777777776));
	} else if (im_m <= 7.2e+239) {
		tmp = Math.log1p(Math.expm1((im_m * -0.004629629629629629)));
	} else if ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = -im_m * (re + (-0.16666666666666666 * Math.pow(re, 3.0)));
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.7e+208:
		tmp = math.log1p(math.expm1((im_m * -0.16666666666666666)))
	elif im_m <= 1.58e+227:
		tmp = math.sqrt((math.pow(im_m, 2.0) * 0.027777777777777776))
	elif im_m <= 7.2e+239:
		tmp = math.log1p(math.expm1((im_m * -0.004629629629629629)))
	elif (im_m <= 6.5e+252) or not (im_m <= 6.6e+273):
		tmp = -im_m * (re + (-0.16666666666666666 * math.pow(re, 3.0)))
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.7e+208)
		tmp = log1p(expm1(Float64(im_m * -0.16666666666666666)));
	elseif (im_m <= 1.58e+227)
		tmp = sqrt(Float64((im_m ^ 2.0) * 0.027777777777777776));
	elseif (im_m <= 7.2e+239)
		tmp = log1p(expm1(Float64(im_m * -0.004629629629629629)));
	elseif ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273))
		tmp = Float64(Float64(-im_m) * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0))));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.7e+208], N[Log[1 + N[(Exp[N[(im$95$m * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im$95$m, 1.58e+227], N[Sqrt[N[(N[Power[im$95$m, 2.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision], If[LessEqual[im$95$m, 7.2e+239], N[Log[1 + N[(Exp[N[(im$95$m * -0.004629629629629629), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im$95$m, 6.5e+252], N[Not[LessEqual[im$95$m, 6.6e+273]], $MachinePrecision]], N[((-im$95$m) * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]

Derivation

1. Split input into 6 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 2.7e208

   1. Initial program 98.3%

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

   3. Taylor expanded in im around 0 72.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 72.5%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 72.5%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 72.5%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 72.5%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 72.5%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 76.6%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 2.5%

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

      1. log1p-expm1-u 51.2%

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

   9. Applied egg-rr 51.2%

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

   if 2.7e208 < im < 1.57999999999999994e227

   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%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 100.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 100.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 3.6%

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

      1. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{im \cdot -0.16666666666666666} \cdot \sqrt{im \cdot -0.16666666666666666}} \]

      2. sqrt-unprod 40.0%

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

      3. swap-sqr 40.0%

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

      4. unpow2 40.0%

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

      5. metadata-eval 40.0%

         \[\leadsto \sqrt{{im}^{2} \cdot \color{blue}{0.027777777777777776}} \]

   9. Applied egg-rr 40.0%

      \[\leadsto \color{blue}{\sqrt{{im}^{2} \cdot 0.027777777777777776}} \]

   if 1.57999999999999994e227 < im < 7.2e239

   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%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 100.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 100.0%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 100.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 4.7%

      \[\leadsto im \cdot \color{blue}{-0.004629629629629629} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 75.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.004629629629629629\right)\right)} \]

   9. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.004629629629629629\right)\right)} \]

   if 7.2e239 < im < 6.5e252 or 6.59999999999999971e273 < 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 8.7%

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

      1. associate-*r* 8.7%

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

      2. neg-mul-1 8.7%

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

   5. Simplified 8.7%

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

   7. Applied egg-rr 8.7%

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

   8. Taylor expanded in re around 0 86.5%

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

      1. distribute-rgt-in 86.5%

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

      2. *-lft-identity 86.5%

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

      3. associate-*l* 86.5%

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

      4. unpow2 86.5%

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

      5. unpow3 86.5%

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

   10. Simplified 86.5%

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

   if 6.5e252 < im < 6.59999999999999971e273

   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. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      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%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. unpow2 100.0%

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

      15. cube-unmult 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

      1. fma-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot re\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;\sqrt{{im}^{2} \cdot 0.027777777777777776}\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.004629629629629629\right)\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+252} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left(-im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 10^{+39}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 8.2e-8)
    (* (- im_m) (sin re))
    (if (<= im_m 1e+39)
      (* (- (exp (- im_m)) (exp im_m)) (* re 0.5))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 8.2e-8) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1e+39) {
		tmp = (exp(-im_m) - exp(im_m)) * (re * 0.5);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	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 <= 8.2d-8) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1d+39) then
        tmp = (exp(-im_m) - exp(im_m)) * (re * 0.5d0)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    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 <= 8.2e-8) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1e+39) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (re * 0.5);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 8.2e-8:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1e+39:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (re * 0.5)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 8.2e-8)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1e+39)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(re * 0.5));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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 <= 8.2e-8)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1e+39)
		tmp = (exp(-im_m) - exp(im_m)) * (re * 0.5);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	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, 8.2e-8], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1e+39], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 8.20000000000000063e-8

   1. Initial program 57.8%

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

   3. Taylor expanded in im around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. neg-mul-1 61.7%

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

   5. Simplified 61.7%

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

   if 8.20000000000000063e-8 < im < 9.9999999999999994e38

   1. Initial program 99.5%

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

   3. Taylor expanded in re around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 9.9999999999999994e38 < 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 91.8%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.8%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.8%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.8%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.8%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.8%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around inf 96.8%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 10^{+39}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.2:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.2)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 2.4e+29)
      (log1p (expm1 (* im_m -0.16666666666666666)))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 0.2) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 2.4e+29) {
		tmp = log1p(expm1((im_m * -0.16666666666666666)));
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.2) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 2.4e+29) {
		tmp = Math.log1p(Math.expm1((im_m * -0.16666666666666666)));
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 0.2:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 2.4e+29:
		tmp = math.log1p(math.expm1((im_m * -0.16666666666666666)))
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 0.2)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 2.4e+29)
		tmp = log1p(expm1(Float64(im_m * -0.16666666666666666)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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, 0.2], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.4e+29], N[Log[1 + N[(Exp[N[(im$95$m * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.20000000000000001

   1. Initial program 58.0%

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

   3. Taylor expanded in im around 0 89.4%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*r* 89.4%

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

      6. distribute-lft-out-- 89.4%

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

      7. associate-*r* 89.4%

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

      8. *-commutative 89.4%

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

      9. associate-*r* 89.4%

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

      10. associate-*r* 92.8%

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

      11. distribute-rgt-out-- 92.8%

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

      12. *-commutative 92.8%

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

      13. associate-*r* 92.8%

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

      14. unpow2 92.8%

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

      15. cube-unmult 92.8%

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

   5. Simplified 92.8%

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

   if 0.20000000000000001 < im < 2.4000000000000001e29

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0 3.6%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 3.6%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 3.6%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 3.6%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 3.6%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 3.6%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 2.2%

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

      1. log1p-expm1-u 57.2%

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

   9. Applied egg-rr 57.2%

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

   if 2.4000000000000001e29 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 90.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 90.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 90.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 90.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 90.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around inf 95.2%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.2:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00078:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00078)
    (* (- im_m) (sin re))
    (if (<= im_m 1e+39)
      (log1p (expm1 (* im_m -0.16666666666666666)))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

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 <= 0.00078) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1e+39) {
		tmp = log1p(expm1((im_m * -0.16666666666666666)));
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00078) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1e+39) {
		tmp = Math.log1p(Math.expm1((im_m * -0.16666666666666666)));
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	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 <= 0.00078:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1e+39:
		tmp = math.log1p(math.expm1((im_m * -0.16666666666666666)))
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	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 <= 0.00078)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1e+39)
		tmp = log1p(expm1(Float64(im_m * -0.16666666666666666)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	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, 0.00078], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1e+39], N[Log[1 + N[(Exp[N[(im$95$m * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7.79999999999999986e-4

   1. Initial program 57.8%

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

   3. Taylor expanded in im around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. neg-mul-1 61.7%

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

   5. Simplified 61.7%

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

   if 7.79999999999999986e-4 < im < 9.9999999999999994e38

   1. Initial program 99.5%

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

   3. Taylor expanded in im around 0 14.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 14.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 14.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 14.0%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 14.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 14.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 14.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 2.2%

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

      1. log1p-expm1-u 55.8%

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

   9. Applied egg-rr 55.8%

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

   if 9.9999999999999994e38 < 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 91.8%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.8%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.8%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.8%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.8%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.8%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   6. Taylor expanded in im around inf 96.8%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00078:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.5 \cdot 10^{+253} \lor \neg \left(im\_m \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left(-im\_m\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.6e-15)
    (* (- im_m) (sin re))
    (if (<= im_m 7.2e+239)
      (log1p (expm1 (* im_m -0.16666666666666666)))
      (if (or (<= im_m 2.5e+253) (not (<= im_m 6.6e+273)))
        (* (- im_m) (+ re (* -0.16666666666666666 (pow re 3.0))))
        (* -0.16666666666666666 (* re (pow im_m 3.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 7.2e+239) {
		tmp = log1p(expm1((im_m * -0.16666666666666666)));
	} else if ((im_m <= 2.5e+253) || !(im_m <= 6.6e+273)) {
		tmp = -im_m * (re + (-0.16666666666666666 * pow(re, 3.0)));
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 7.2e+239) {
		tmp = Math.log1p(Math.expm1((im_m * -0.16666666666666666)));
	} else if ((im_m <= 2.5e+253) || !(im_m <= 6.6e+273)) {
		tmp = -im_m * (re + (-0.16666666666666666 * Math.pow(re, 3.0)));
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 7.2e+239:
		tmp = math.log1p(math.expm1((im_m * -0.16666666666666666)))
	elif (im_m <= 2.5e+253) or not (im_m <= 6.6e+273):
		tmp = -im_m * (re + (-0.16666666666666666 * math.pow(re, 3.0)))
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 7.2e+239)
		tmp = log1p(expm1(Float64(im_m * -0.16666666666666666)));
	elseif ((im_m <= 2.5e+253) || !(im_m <= 6.6e+273))
		tmp = Float64(Float64(-im_m) * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0))));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.2e+239], N[Log[1 + N[(Exp[N[(im$95$m * -0.16666666666666666), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im$95$m, 2.5e+253], N[Not[LessEqual[im$95$m, 6.6e+273]], $MachinePrecision]], N[((-im$95$m) * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 7.2e239

   1. Initial program 98.6%

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

   3. Taylor expanded in im around 0 77.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 77.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 77.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 77.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 77.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 77.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 80.5%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 2.8%

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

      1. log1p-expm1-u 53.8%

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

   9. Applied egg-rr 53.8%

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

   if 7.2e239 < im < 2.4999999999999998e253 or 6.59999999999999971e273 < 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 8.7%

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

      1. associate-*r* 8.7%

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

      2. neg-mul-1 8.7%

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

   5. Simplified 8.7%

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

   7. Applied egg-rr 8.7%

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

   8. Taylor expanded in re around 0 86.5%

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

      1. distribute-rgt-in 86.5%

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

      2. *-lft-identity 86.5%

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

      3. associate-*l* 86.5%

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

      4. unpow2 86.5%

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

      5. unpow3 86.5%

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

   10. Simplified 86.5%

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

   if 2.4999999999999998e253 < im < 6.59999999999999971e273

   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. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      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%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. unpow2 100.0%

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

      15. cube-unmult 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

      1. fma-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+253} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left(-im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(re \cdot {im\_m}^{5}\right)\\ t_1 := \left(-im\_m\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.5 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im\_m \leq 6.5 \cdot 10^{+239}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 7.5 \cdot 10^{+252} \lor \neg \left(im\_m \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* re (pow im_m 5.0))))
        (t_1 (* (- im_m) (+ re (* -0.16666666666666666 (pow re 3.0))))))
   (*
    im_s
    (if (<= im_m 2.6e-15)
      (* (- im_m) (sin re))
      (if (<= im_m 8.2e+26)
        (* im_m (- (expm1 re)))
        (if (<= im_m 2.5e+208)
          t_0
          (if (<= im_m 1.58e+227)
            t_1
            (if (<= im_m 6.5e+239)
              t_0
              (if (or (<= im_m 7.5e+252) (not (<= im_m 6.6e+273)))
                t_1
                (* -0.16666666666666666 (* re (pow im_m 3.0))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * pow(im_m, 5.0));
	double t_1 = -im_m * (re + (-0.16666666666666666 * pow(re, 3.0)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -expm1(re);
	} else if (im_m <= 2.5e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 6.5e+239) {
		tmp = t_0;
	} else if ((im_m <= 7.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * Math.pow(im_m, 5.0));
	double t_1 = -im_m * (re + (-0.16666666666666666 * Math.pow(re, 3.0)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -Math.expm1(re);
	} else if (im_m <= 2.5e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 6.5e+239) {
		tmp = t_0;
	} else if ((im_m <= 7.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -0.008333333333333333 * (re * math.pow(im_m, 5.0))
	t_1 = -im_m * (re + (-0.16666666666666666 * math.pow(re, 3.0)))
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 8.2e+26:
		tmp = im_m * -math.expm1(re)
	elif im_m <= 2.5e+208:
		tmp = t_0
	elif im_m <= 1.58e+227:
		tmp = t_1
	elif im_m <= 6.5e+239:
		tmp = t_0
	elif (im_m <= 7.5e+252) or not (im_m <= 6.6e+273):
		tmp = t_1
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-0.008333333333333333 * Float64(re * (im_m ^ 5.0)))
	t_1 = Float64(Float64(-im_m) * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0))))
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 8.2e+26)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif (im_m <= 2.5e+208)
		tmp = t_0;
	elseif (im_m <= 1.58e+227)
		tmp = t_1;
	elseif (im_m <= 6.5e+239)
		tmp = t_0;
	elseif ((im_m <= 7.5e+252) || !(im_m <= 6.6e+273))
		tmp = t_1;
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(re * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-im$95$m) * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 8.2e+26], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.5e+208], t$95$0, If[LessEqual[im$95$m, 1.58e+227], t$95$1, If[LessEqual[im$95$m, 6.5e+239], t$95$0, If[Or[LessEqual[im$95$m, 7.5e+252], N[Not[LessEqual[im$95$m, 6.6e+273]], $MachinePrecision]], t$95$1, N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 8.19999999999999967e26

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 8.19999999999999967e26 < im < 2.5000000000000002e208 or 1.57999999999999994e227 < im < 6.5e239

   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 re around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. unsub-neg 50.9%

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

      4. associate-*r* 50.9%

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

      5. distribute-rgt-out 50.9%

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

      6. *-commutative 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in im around inf 55.3%

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

   if 2.5000000000000002e208 < im < 1.57999999999999994e227 or 6.5e239 < im < 7.4999999999999995e252 or 6.59999999999999971e273 < 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 7.1%

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

      1. associate-*r* 7.1%

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

      2. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 7.1%

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

   8. Taylor expanded in re around 0 76.1%

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

      1. distribute-rgt-in 76.1%

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

      2. *-lft-identity 76.1%

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

      3. associate-*l* 76.1%

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

      4. unpow2 76.1%

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

      5. unpow3 76.1%

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

   10. Simplified 76.1%

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

   if 7.4999999999999995e252 < im < 6.59999999999999971e273

   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. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      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%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. unpow2 100.0%

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

      15. cube-unmult 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

      1. fma-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+208}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;\left(-im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+239}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+252} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left(-im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(re \cdot {im\_m}^{5}\right)\\ t_1 := im\_m \cdot \left({re}^{3} \cdot 0.16666666666666666\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.65 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+253} \lor \neg \left(im\_m \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* re (pow im_m 5.0))))
        (t_1 (* im_m (* (pow re 3.0) 0.16666666666666666))))
   (*
    im_s
    (if (<= im_m 2.6e-15)
      (* (- im_m) (sin re))
      (if (<= im_m 2.7e+26)
        (* im_m (- (expm1 re)))
        (if (<= im_m 2e+208)
          t_0
          (if (<= im_m 1.65e+227)
            t_1
            (if (<= im_m 7.2e+239)
              t_0
              (if (or (<= im_m 9.6e+253) (not (<= im_m 6.6e+273)))
                t_1
                (* -0.16666666666666666 (* re (pow im_m 3.0))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * pow(im_m, 5.0));
	double t_1 = im_m * (pow(re, 3.0) * 0.16666666666666666);
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 2.7e+26) {
		tmp = im_m * -expm1(re);
	} else if (im_m <= 2e+208) {
		tmp = t_0;
	} else if (im_m <= 1.65e+227) {
		tmp = t_1;
	} else if (im_m <= 7.2e+239) {
		tmp = t_0;
	} else if ((im_m <= 9.6e+253) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * Math.pow(im_m, 5.0));
	double t_1 = im_m * (Math.pow(re, 3.0) * 0.16666666666666666);
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 2.7e+26) {
		tmp = im_m * -Math.expm1(re);
	} else if (im_m <= 2e+208) {
		tmp = t_0;
	} else if (im_m <= 1.65e+227) {
		tmp = t_1;
	} else if (im_m <= 7.2e+239) {
		tmp = t_0;
	} else if ((im_m <= 9.6e+253) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -0.008333333333333333 * (re * math.pow(im_m, 5.0))
	t_1 = im_m * (math.pow(re, 3.0) * 0.16666666666666666)
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 2.7e+26:
		tmp = im_m * -math.expm1(re)
	elif im_m <= 2e+208:
		tmp = t_0
	elif im_m <= 1.65e+227:
		tmp = t_1
	elif im_m <= 7.2e+239:
		tmp = t_0
	elif (im_m <= 9.6e+253) or not (im_m <= 6.6e+273):
		tmp = t_1
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-0.008333333333333333 * Float64(re * (im_m ^ 5.0)))
	t_1 = Float64(im_m * Float64((re ^ 3.0) * 0.16666666666666666))
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 2.7e+26)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif (im_m <= 2e+208)
		tmp = t_0;
	elseif (im_m <= 1.65e+227)
		tmp = t_1;
	elseif (im_m <= 7.2e+239)
		tmp = t_0;
	elseif ((im_m <= 9.6e+253) || !(im_m <= 6.6e+273))
		tmp = t_1;
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(re * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im$95$m * N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.7e+26], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2e+208], t$95$0, If[LessEqual[im$95$m, 1.65e+227], t$95$1, If[LessEqual[im$95$m, 7.2e+239], t$95$0, If[Or[LessEqual[im$95$m, 9.6e+253], N[Not[LessEqual[im$95$m, 6.6e+273]], $MachinePrecision]], t$95$1, N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 2.7e26

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 2.7e26 < im < 2e208 or 1.6499999999999999e227 < im < 7.2e239

   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 re around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. unsub-neg 50.9%

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

      4. associate-*r* 50.9%

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

      5. distribute-rgt-out 50.9%

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

      6. *-commutative 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in im around inf 55.3%

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

   if 2e208 < im < 1.6499999999999999e227 or 7.2e239 < im < 9.59999999999999965e253 or 6.59999999999999971e273 < 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 7.1%

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

      1. associate-*r* 7.1%

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

      2. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 7.1%

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

   8. Taylor expanded in re around 0 76.1%

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

      1. distribute-rgt-in 76.1%

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

      2. *-lft-identity 76.1%

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

      3. associate-*l* 76.1%

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

      4. unpow2 76.1%

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

      5. unpow3 76.1%

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

   10. Simplified 76.1%

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

   11. Taylor expanded in re around inf 75.3%

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

       1. *-commutative 75.3%

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

       2. associate-*l* 75.3%

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

   13. Simplified 75.3%

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

   if 9.59999999999999965e253 < im < 6.59999999999999971e273

   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. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      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%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. unpow2 100.0%

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

      15. cube-unmult 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

      1. fma-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+208}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+227}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+253} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot \left(re \cdot {im\_m}^{5}\right)\\ t_1 := \left|im\_m \cdot \left(-4 - re\right)\right|\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im\_m \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 6.5 \cdot 10^{+252} \lor \neg \left(im\_m \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (* re (pow im_m 5.0))))
        (t_1 (fabs (* im_m (- -4.0 re)))))
   (*
    im_s
    (if (<= im_m 2.6e-15)
      (* (- im_m) (sin re))
      (if (<= im_m 8.5e+25)
        (* im_m (- (expm1 re)))
        (if (<= im_m 2.7e+208)
          t_0
          (if (<= im_m 1.58e+227)
            t_1
            (if (<= im_m 7.2e+239)
              t_0
              (if (or (<= im_m 6.5e+252) (not (<= im_m 6.6e+273)))
                t_1
                (* -0.16666666666666666 (* re (pow im_m 3.0))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * pow(im_m, 5.0));
	double t_1 = fabs((im_m * (-4.0 - re)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 8.5e+25) {
		tmp = im_m * -expm1(re);
	} else if (im_m <= 2.7e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 7.2e+239) {
		tmp = t_0;
	} else if ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -0.008333333333333333 * (re * Math.pow(im_m, 5.0));
	double t_1 = Math.abs((im_m * (-4.0 - re)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 8.5e+25) {
		tmp = im_m * -Math.expm1(re);
	} else if (im_m <= 2.7e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 7.2e+239) {
		tmp = t_0;
	} else if ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273)) {
		tmp = t_1;
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -0.008333333333333333 * (re * math.pow(im_m, 5.0))
	t_1 = math.fabs((im_m * (-4.0 - re)))
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 8.5e+25:
		tmp = im_m * -math.expm1(re)
	elif im_m <= 2.7e+208:
		tmp = t_0
	elif im_m <= 1.58e+227:
		tmp = t_1
	elif im_m <= 7.2e+239:
		tmp = t_0
	elif (im_m <= 6.5e+252) or not (im_m <= 6.6e+273):
		tmp = t_1
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-0.008333333333333333 * Float64(re * (im_m ^ 5.0)))
	t_1 = abs(Float64(im_m * Float64(-4.0 - re)))
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 8.5e+25)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif (im_m <= 2.7e+208)
		tmp = t_0;
	elseif (im_m <= 1.58e+227)
		tmp = t_1;
	elseif (im_m <= 7.2e+239)
		tmp = t_0;
	elseif ((im_m <= 6.5e+252) || !(im_m <= 6.6e+273))
		tmp = t_1;
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[(re * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(im$95$m * N[(-4.0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 8.5e+25], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.7e+208], t$95$0, If[LessEqual[im$95$m, 1.58e+227], t$95$1, If[LessEqual[im$95$m, 7.2e+239], t$95$0, If[Or[LessEqual[im$95$m, 6.5e+252], N[Not[LessEqual[im$95$m, 6.6e+273]], $MachinePrecision]], t$95$1, N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 8.5000000000000007e25

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 8.5000000000000007e25 < im < 2.7e208 or 1.57999999999999994e227 < im < 7.2e239

   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 re around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-commutative 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in im around 0 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. unsub-neg 50.9%

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

      4. associate-*r* 50.9%

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

      5. distribute-rgt-out 50.9%

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

      6. *-commutative 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in im around inf 55.3%

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

   if 2.7e208 < im < 1.57999999999999994e227 or 7.2e239 < im < 6.5e252 or 6.59999999999999971e273 < 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 7.1%

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

      1. associate-*r* 7.1%

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

      2. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 4.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 4.7%

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

      2. rem-exp-log 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. metadata-eval 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in re around 0 1.1%

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

       1. *-commutative 1.1%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 1.1%

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

       3. distribute-rgt-neg-out 1.1%

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

       4. distribute-lft-out 1.1%

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

       5. unsub-neg 1.1%

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

   12. Simplified 1.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 41.7%

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

       3. pow2 41.7%

          \[\leadsto \sqrt{\color{blue}{{\left(im \cdot \left(-4 - re\right)\right)}^{2}}} \]

   14. Applied egg-rr 41.7%

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

       1. unpow2 41.7%

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

       2. rem-sqrt-square 33.8%

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

   16. Simplified 33.8%

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

   if 6.5e252 < im < 6.59999999999999971e273

   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. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-lft-out-- 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      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%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. unpow2 100.0%

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

      15. cube-unmult 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 100.0%

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

      1. fma-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;\left|im \cdot \left(-4 - re\right)\right|\\ \mathbf{elif}\;im \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+252} \lor \neg \left(im \leq 6.6 \cdot 10^{+273}\right):\\ \;\;\;\;\left|im \cdot \left(-4 - re\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(re \cdot {im\_m}^{3}\right)\\ t_1 := \left|im\_m \cdot \left(-4 - re\right)\right|\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im\_m \leq 6.4 \cdot 10^{+239}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \frac{-4}{re} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 6.5 \cdot 10^{+252} \lor \neg \left(im\_m \leq 8.5 \cdot 10^{+272}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* re (pow im_m 3.0))))
        (t_1 (fabs (* im_m (- -4.0 re)))))
   (*
    im_s
    (if (<= im_m 2.6e-15)
      (* (- im_m) (sin re))
      (if (<= im_m 8.2e+26)
        (* im_m (- (expm1 re)))
        (if (<= im_m 2.7e+208)
          t_0
          (if (<= im_m 1.58e+227)
            t_1
            (if (<= im_m 6.4e+239)
              (* re (- (* im_m (/ -4.0 re)) im_m))
              (if (or (<= im_m 6.5e+252) (not (<= im_m 8.5e+272)))
                t_1
                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 = -0.16666666666666666 * (re * pow(im_m, 3.0));
	double t_1 = fabs((im_m * (-4.0 - re)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -expm1(re);
	} else if (im_m <= 2.7e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 6.4e+239) {
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	} else if ((im_m <= 6.5e+252) || !(im_m <= 8.5e+272)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	double t_1 = Math.abs((im_m * (-4.0 - re)));
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -Math.expm1(re);
	} else if (im_m <= 2.7e+208) {
		tmp = t_0;
	} else if (im_m <= 1.58e+227) {
		tmp = t_1;
	} else if (im_m <= 6.4e+239) {
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	} else if ((im_m <= 6.5e+252) || !(im_m <= 8.5e+272)) {
		tmp = t_1;
	} else {
		tmp = 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 = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	t_1 = math.fabs((im_m * (-4.0 - re)))
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 8.2e+26:
		tmp = im_m * -math.expm1(re)
	elif im_m <= 2.7e+208:
		tmp = t_0
	elif im_m <= 1.58e+227:
		tmp = t_1
	elif im_m <= 6.4e+239:
		tmp = re * ((im_m * (-4.0 / re)) - im_m)
	elif (im_m <= 6.5e+252) or not (im_m <= 8.5e+272):
		tmp = t_1
	else:
		tmp = 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(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)))
	t_1 = abs(Float64(im_m * Float64(-4.0 - re)))
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 8.2e+26)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif (im_m <= 2.7e+208)
		tmp = t_0;
	elseif (im_m <= 1.58e+227)
		tmp = t_1;
	elseif (im_m <= 6.4e+239)
		tmp = Float64(re * Float64(Float64(im_m * Float64(-4.0 / re)) - im_m));
	elseif ((im_m <= 6.5e+252) || !(im_m <= 8.5e+272))
		tmp = t_1;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(im$95$m * N[(-4.0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 8.2e+26], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.7e+208], t$95$0, If[LessEqual[im$95$m, 1.58e+227], t$95$1, If[LessEqual[im$95$m, 6.4e+239], N[(re * N[(N[(im$95$m * N[(-4.0 / re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 6.5e+252], N[Not[LessEqual[im$95$m, 8.5e+272]], $MachinePrecision]], t$95$1, t$95$0]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 8.19999999999999967e26

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 8.19999999999999967e26 < im < 2.7e208 or 6.5e252 < im < 8.49999999999999996e272

   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 63.2%

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

      1. +-commutative 63.2%

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

      2. mul-1-neg 63.2%

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

      3. unsub-neg 63.2%

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

      4. *-commutative 63.2%

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

      5. associate-*r* 63.2%

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

      6. distribute-lft-out-- 63.2%

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

      7. associate-*r* 63.2%

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

      8. *-commutative 63.2%

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

      9. associate-*r* 63.2%

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

      10. associate-*r* 69.6%

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

      11. distribute-rgt-out-- 69.6%

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

      12. *-commutative 69.6%

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

      13. associate-*r* 69.6%

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

      14. unpow2 69.6%

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

      15. cube-unmult 69.6%

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

   5. Simplified 69.6%

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

   6. Taylor expanded in re around 0 52.8%

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

      1. fma-neg 52.8%

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

   8. Simplified 52.8%

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

   9. Taylor expanded in im around inf 52.8%

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

   if 2.7e208 < im < 1.57999999999999994e227 or 6.4000000000000003e239 < im < 6.5e252 or 8.49999999999999996e272 < 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 7.1%

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

      1. associate-*r* 7.1%

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

      2. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 4.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 4.7%

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

      2. rem-exp-log 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. metadata-eval 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in re around 0 1.1%

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

       1. *-commutative 1.1%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 1.1%

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

       3. distribute-rgt-neg-out 1.1%

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

       4. distribute-lft-out 1.1%

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

       5. unsub-neg 1.1%

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

   12. Simplified 1.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 41.7%

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

       3. pow2 41.7%

          \[\leadsto \sqrt{\color{blue}{{\left(im \cdot \left(-4 - re\right)\right)}^{2}}} \]

   14. Applied egg-rr 41.7%

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

       1. unpow2 41.7%

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

       2. rem-sqrt-square 33.8%

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

   16. Simplified 33.8%

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

   if 1.57999999999999994e227 < im < 6.4000000000000003e239

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   5. Simplified 4.2%

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

   7. Applied egg-rr 4.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 4.7%

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

      2. rem-exp-log 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. metadata-eval 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in re around 0 29.7%

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

       1. *-commutative 29.7%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 29.7%

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

       3. distribute-rgt-neg-out 29.7%

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

       4. distribute-lft-out 29.7%

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

       5. unsub-neg 29.7%

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

   12. Simplified 29.7%

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

   13. Taylor expanded in re around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. unsub-neg 100.0%

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

       3. associate-*r/ 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-/l* 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 1.58 \cdot 10^{+227}:\\ \;\;\;\;\left|im \cdot \left(-4 - re\right)\right|\\ \mathbf{elif}\;im \leq 6.4 \cdot 10^{+239}:\\ \;\;\;\;re \cdot \left(im \cdot \frac{-4}{re} - im\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+252} \lor \neg \left(im \leq 8.5 \cdot 10^{+272}\right):\\ \;\;\;\;\left|im \cdot \left(-4 - re\right)\right|\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.7 \cdot 10^{+208} \lor \neg \left(im\_m \leq 1.58 \cdot 10^{+227}\right) \land \left(im\_m \leq 7.2 \cdot 10^{+239} \lor \neg \left(im\_m \leq 6.5 \cdot 10^{+252}\right) \land im\_m \leq 3.9 \cdot 10^{+273}\right):\\ \;\;\;\;re \cdot \left(im\_m \cdot \frac{-4}{re} - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im\_m \cdot \left(-4 - re\right)\right|\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.6e-15)
    (* (- im_m) (sin re))
    (if (<= im_m 8.2e+26)
      (* im_m (- (expm1 re)))
      (if (or (<= im_m 2.7e+208)
              (and (not (<= im_m 1.58e+227))
                   (or (<= im_m 7.2e+239)
                       (and (not (<= im_m 6.5e+252)) (<= im_m 3.9e+273)))))
        (* re (- (* im_m (/ -4.0 re)) im_m))
        (fabs (* im_m (- -4.0 re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -expm1(re);
	} else if ((im_m <= 2.7e+208) || (!(im_m <= 1.58e+227) && ((im_m <= 7.2e+239) || (!(im_m <= 6.5e+252) && (im_m <= 3.9e+273))))) {
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	} else {
		tmp = fabs((im_m * (-4.0 - re)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 8.2e+26) {
		tmp = im_m * -Math.expm1(re);
	} else if ((im_m <= 2.7e+208) || (!(im_m <= 1.58e+227) && ((im_m <= 7.2e+239) || (!(im_m <= 6.5e+252) && (im_m <= 3.9e+273))))) {
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	} else {
		tmp = Math.abs((im_m * (-4.0 - re)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 8.2e+26:
		tmp = im_m * -math.expm1(re)
	elif (im_m <= 2.7e+208) or (not (im_m <= 1.58e+227) and ((im_m <= 7.2e+239) or (not (im_m <= 6.5e+252) and (im_m <= 3.9e+273)))):
		tmp = re * ((im_m * (-4.0 / re)) - im_m)
	else:
		tmp = math.fabs((im_m * (-4.0 - re)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 8.2e+26)
		tmp = Float64(im_m * Float64(-expm1(re)));
	elseif ((im_m <= 2.7e+208) || (!(im_m <= 1.58e+227) && ((im_m <= 7.2e+239) || (!(im_m <= 6.5e+252) && (im_m <= 3.9e+273)))))
		tmp = Float64(re * Float64(Float64(im_m * Float64(-4.0 / re)) - im_m));
	else
		tmp = abs(Float64(im_m * Float64(-4.0 - re)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 8.2e+26], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 2.7e+208], And[N[Not[LessEqual[im$95$m, 1.58e+227]], $MachinePrecision], Or[LessEqual[im$95$m, 7.2e+239], And[N[Not[LessEqual[im$95$m, 6.5e+252]], $MachinePrecision], LessEqual[im$95$m, 3.9e+273]]]]], N[(re * N[(N[(im$95$m * N[(-4.0 / re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[Abs[N[(im$95$m * N[(-4.0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 8.19999999999999967e26

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 8.19999999999999967e26 < im < 2.7e208 or 1.57999999999999994e227 < im < 7.2e239 or 6.5e252 < im < 3.9000000000000001e273

   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 3.9%

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

      1. associate-*r* 3.9%

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

      2. neg-mul-1 3.9%

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

   5. Simplified 3.9%

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

   7. Applied egg-rr 3.1%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.1%

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

      2. rem-exp-log 3.1%

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

      3. +-commutative 3.1%

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

      4. associate--l+ 3.1%

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

      5. metadata-eval 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in re around 0 12.9%

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

       1. *-commutative 12.9%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 12.9%

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

       3. distribute-rgt-neg-out 12.9%

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

       4. distribute-lft-out 12.9%

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

       5. unsub-neg 12.9%

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

   12. Simplified 12.9%

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

   13. Taylor expanded in re around inf 36.3%

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

       1. mul-1-neg 36.3%

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

       2. unsub-neg 36.3%

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

       3. associate-*r/ 36.3%

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

       4. *-commutative 36.3%

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

       5. associate-/l* 36.3%

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

   15. Simplified 36.3%

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

   if 2.7e208 < im < 1.57999999999999994e227 or 7.2e239 < im < 6.5e252 or 3.9000000000000001e273 < 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 7.1%

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

      1. associate-*r* 7.1%

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

      2. neg-mul-1 7.1%

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 4.7%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 4.7%

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

      2. rem-exp-log 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. metadata-eval 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in re around 0 1.1%

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

       1. *-commutative 1.1%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 1.1%

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

       3. distribute-rgt-neg-out 1.1%

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

       4. distribute-lft-out 1.1%

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

       5. unsub-neg 1.1%

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

   12. Simplified 1.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 41.7%

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

       3. pow2 41.7%

          \[\leadsto \sqrt{\color{blue}{{\left(im \cdot \left(-4 - re\right)\right)}^{2}}} \]

   14. Applied egg-rr 41.7%

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

       1. unpow2 41.7%

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

       2. rem-sqrt-square 33.8%

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

   16. Simplified 33.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+208} \lor \neg \left(im \leq 1.58 \cdot 10^{+227}\right) \land \left(im \leq 7.2 \cdot 10^{+239} \lor \neg \left(im \leq 6.5 \cdot 10^{+252}\right) \land im \leq 3.9 \cdot 10^{+273}\right):\\ \;\;\;\;re \cdot \left(im \cdot \frac{-4}{re} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left|im \cdot \left(-4 - re\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \frac{-4}{re} - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.6e-15)
    (* (- im_m) (sin re))
    (if (<= im_m 1.1e+23)
      (* im_m (- (expm1 re)))
      (* re (- (* im_m (/ -4.0 re)) 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 <= 2.6e-15) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.1e+23) {
		tmp = im_m * -expm1(re);
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 2.6e-15) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.1e+23) {
		tmp = im_m * -Math.expm1(re);
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 2.6e-15:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.1e+23:
		tmp = im_m * -math.expm1(re)
	else:
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 2.6e-15)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.1e+23)
		tmp = Float64(im_m * Float64(-expm1(re)));
	else
		tmp = Float64(re * Float64(Float64(im_m * Float64(-4.0 / re)) - 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, 2.6e-15], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.1e+23], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(im$95$m * N[(-4.0 / re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.60000000000000004e-15

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. neg-mul-1 61.5%

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

   5. Simplified 61.5%

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

   if 2.60000000000000004e-15 < im < 1.10000000000000004e23

   1. Initial program 89.1%

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

   3. Taylor expanded in im around 0 21.3%

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

      1. associate-*r* 21.3%

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

      2. neg-mul-1 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in re around 0 30.3%

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

   if 1.10000000000000004e23 < 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 4.6%

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

      1. associate-*r* 4.6%

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

      2. neg-mul-1 4.6%

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

   5. Simplified 4.6%

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

   7. Applied egg-rr 3.4%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.4%

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

      2. rem-exp-log 3.4%

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

      3. +-commutative 3.4%

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

      4. associate--l+ 3.4%

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

      5. metadata-eval 3.4%

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

   9. Simplified 3.4%

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

   10. Taylor expanded in re around 0 10.5%

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

       1. *-commutative 10.5%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 10.5%

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

       3. distribute-rgt-neg-out 10.5%

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

       4. distribute-lft-out 10.5%

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

       5. unsub-neg 10.5%

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

   12. Simplified 10.5%

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

   13. Taylor expanded in re around inf 30.8%

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

       1. mul-1-neg 30.8%

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

       2. unsub-neg 30.8%

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

       3. associate-*r/ 30.8%

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

       4. *-commutative 30.8%

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

       5. associate-/l* 30.8%

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

   15. Simplified 30.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \frac{-4}{re} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.05 \cdot 10^{+109}:\\ \;\;\;\;im\_m \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \frac{-4}{re} - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.05e+109)
    (* im_m (- (expm1 re)))
    (* re (- (* im_m (/ -4.0 re)) 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 <= 1.05e+109) {
		tmp = im_m * -expm1(re);
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 1.05e+109) {
		tmp = im_m * -Math.expm1(re);
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 1.05e+109:
		tmp = im_m * -math.expm1(re)
	else:
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 1.05e+109)
		tmp = Float64(im_m * Float64(-expm1(re)));
	else
		tmp = Float64(re * Float64(Float64(im_m * Float64(-4.0 / re)) - 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, 1.05e+109], N[(im$95$m * (-N[(Exp[re] - 1), $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(im$95$m * N[(-4.0 / re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.0500000000000001e109

   1. Initial program 61.8%

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

   3. Taylor expanded in im around 0 56.3%

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

      1. associate-*r* 56.3%

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

      2. neg-mul-1 56.3%

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

   5. Simplified 56.3%

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

   7. Applied egg-rr 56.2%

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

   8. Taylor expanded in re around 0 33.3%

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

   if 1.0500000000000001e109 < 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 5.0%

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

      1. associate-*r* 5.0%

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

      2. neg-mul-1 5.0%

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

   5. Simplified 5.0%

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

   7. Applied egg-rr 3.8%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.8%

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

      2. rem-exp-log 3.8%

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

      3. +-commutative 3.8%

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

      4. associate--l+ 3.8%

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

      5. metadata-eval 3.8%

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

   9. Simplified 3.8%

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

   10. Taylor expanded in re around 0 13.3%

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

       1. *-commutative 13.3%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 13.3%

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

       3. distribute-rgt-neg-out 13.3%

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

       4. distribute-lft-out 13.3%

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

       5. unsub-neg 13.3%

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

   12. Simplified 13.3%

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

   13. Taylor expanded in re around inf 39.8%

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

       1. mul-1-neg 39.8%

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

       2. unsub-neg 39.8%

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

       3. associate-*r/ 39.8%

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

       4. *-commutative 39.8%

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

       5. associate-/l* 39.8%

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

   15. Simplified 39.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{+109}:\\ \;\;\;\;im \cdot \left(-\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \frac{-4}{re} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.3% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00078:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \frac{-4}{re} - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00078) (* im_m (- re)) (* re (- (* im_m (/ -4.0 re)) 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 <= 0.00078) {
		tmp = im_m * -re;
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.00078d0) then
        tmp = im_m * -re
    else
        tmp = re * ((im_m * ((-4.0d0) / re)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00078) {
		tmp = im_m * -re;
	} else {
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 0.00078:
		tmp = im_m * -re
	else:
		tmp = re * ((im_m * (-4.0 / re)) - 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 <= 0.00078)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(re * Float64(Float64(im_m * Float64(-4.0 / re)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.00078)
		tmp = im_m * -re;
	else
		tmp = re * ((im_m * (-4.0 / re)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.00078], N[(im$95$m * (-re)), $MachinePrecision], N[(re * N[(N[(im$95$m * N[(-4.0 / re), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 7.79999999999999986e-4

   1. Initial program 57.8%

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

   3. Taylor expanded in im around 0 61.7%

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

      1. associate-*r* 61.7%

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

      2. neg-mul-1 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in re around 0 35.7%

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

      1. associate-*r* 35.7%

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

      2. neg-mul-1 35.7%

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

   8. Simplified 35.7%

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

   if 7.79999999999999986e-4 < im

   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 im around 0 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

   5. Simplified 4.9%

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

   7. Applied egg-rr 3.2%

      \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
   8. Step-by-step derivation

      1. log1p-undefine 3.2%

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

      2. rem-exp-log 3.2%

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

      3. +-commutative 3.2%

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

      4. associate--l+ 3.2%

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

      5. metadata-eval 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in re around 0 9.9%

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

       1. *-commutative 9.9%

          \[\leadsto \color{blue}{im \cdot -4} + -1 \cdot \left(im \cdot re\right) \]

       2. mul-1-neg 9.9%

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

       3. distribute-rgt-neg-out 9.9%

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

       4. distribute-lft-out 9.9%

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

       5. unsub-neg 9.9%

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

   12. Simplified 9.9%

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

   13. Taylor expanded in re around inf 28.3%

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

       1. mul-1-neg 28.3%

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

       2. unsub-neg 28.3%

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

       3. associate-*r/ 28.3%

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

       4. *-commutative 28.3%

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

       5. associate-/l* 28.3%

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

   15. Simplified 28.3%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00078:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \frac{-4}{re} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 15.9% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 3.4e-83) (* 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 <= 3.4e-83) {
		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 <= 3.4d-83) 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 <= 3.4e-83) {
		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 <= 3.4e-83:
		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 <= 3.4e-83)
		tmp = Float64(im_m * 0.0);
	else
		tmp = 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 <= 3.4e-83)
		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, 3.4e-83], N[(im$95$m * 0.0), $MachinePrecision], im$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 3.3999999999999998e-83

   1. Initial program 72.5%

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

   3. Taylor expanded in im around 0 91.6%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.6%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.6%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.6%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.6%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.6%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.8%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.5%

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

   if 3.3999999999999998e-83 < re

   1. Initial program 60.3%

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

   3. Taylor expanded in im around 0 95.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 95.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 95.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 95.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 95.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 95.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 8.3%

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

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.4 \cdot 10^{-83}:\\ \;\;\;\;im \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 16.0% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{-77}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.9916666666666667\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 7e-77) (* im_m 0.0) (* im_m 0.9916666666666667))))

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 <= 7e-77) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.9916666666666667;
	}
	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 <= 7d-77) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.9916666666666667d0
    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 <= 7e-77) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.9916666666666667;
	}
	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 <= 7e-77:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.9916666666666667
	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 <= 7e-77)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.9916666666666667);
	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 <= 7e-77)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.9916666666666667;
	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, 7e-77], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.9916666666666667), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 7.00000000000000026e-77

   1. Initial program 72.5%

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

   3. Taylor expanded in im around 0 91.6%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.6%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.6%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.6%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.6%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.6%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.8%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.5%

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

   if 7.00000000000000026e-77 < re

   1. Initial program 60.3%

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

   3. Taylor expanded in im around 0 95.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 95.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 95.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 95.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 95.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 95.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 8.0%

      \[\leadsto im \cdot \color{blue}{0.9916666666666667} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 16.1% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 2.05 \cdot 10^{-48}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.75\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 2.05e-48) (* 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 <= 2.05e-48) {
		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 <= 2.05d-48) 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 <= 2.05e-48) {
		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 <= 2.05e-48:
		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 <= 2.05e-48)
		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 <= 2.05e-48)
		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, 2.05e-48], 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 < 2.05000000000000007e-48

   1. Initial program 72.6%

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

   3. Taylor expanded in im around 0 91.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.2%

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

   if 2.05000000000000007e-48 < re

   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 96.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 96.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 96.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 96.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 96.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 96.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 8.0%

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

Alternative 18: 16.1% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 5.8e-45) (* 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 <= 5.8e-45) {
		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 <= 5.8d-45) 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 <= 5.8e-45) {
		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 <= 5.8e-45:
		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 <= 5.8e-45)
		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 <= 5.8e-45)
		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, 5.8e-45], 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 < 5.8e-45

   1. Initial program 72.6%

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

   3. Taylor expanded in im around 0 91.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.2%

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

   if 5.8e-45 < re

   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 96.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 96.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 96.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 96.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 96.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 96.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 7.7%

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

Alternative 19: 16.0% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 5.8e-45) (* 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 <= 5.8e-45) {
		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 <= 5.8d-45) 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 <= 5.8e-45) {
		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 <= 5.8e-45:
		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 <= 5.8e-45)
		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 <= 5.8e-45)
		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, 5.8e-45], 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 < 5.8e-45

   1. Initial program 72.6%

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

   3. Taylor expanded in im around 0 91.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.2%

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

   if 5.8e-45 < re

   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 96.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 96.1%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 96.1%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 96.1%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 96.1%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 96.1%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 7.5%

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

Alternative 20: 16.0% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.25\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 2.5e-43) (* im_m 0.0) (* im_m 0.25))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 2.5e-43) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.25;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 2.5d-43) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.25d0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 2.5e-43) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.25;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 2.5e-43:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.25
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 2.5e-43)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.25);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 2.5e-43)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.25;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 2.5e-43], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.25), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 2.50000000000000009e-43

   1. Initial program 72.7%

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

   3. Taylor expanded in im around 0 91.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.7%

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

   if 2.50000000000000009e-43 < re

   1. Initial program 58.9%

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

   3. Taylor expanded in im around 0 96.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 96.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 96.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 96.0%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 96.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 96.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 7.5%

      \[\leadsto im \cdot \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 16.0% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 2.5e-43) (* im_m 0.0) (* 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 (re <= 2.5e-43) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 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 (re <= 2.5d-43) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 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 (re <= 2.5e-43) {
		tmp = im_m * 0.0;
	} else {
		tmp = 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 re <= 2.5e-43:
		tmp = im_m * 0.0
	else:
		tmp = 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 (re <= 2.5e-43)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 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 (re <= 2.5e-43)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 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[re, 2.5e-43], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 2.50000000000000009e-43

   1. Initial program 72.7%

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

   3. Taylor expanded in im around 0 91.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.3%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.3%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.3%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.7%

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

   if 2.50000000000000009e-43 < re

   1. Initial program 58.9%

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

   3. Taylor expanded in im around 0 96.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 96.0%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 96.0%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 96.0%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 96.0%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 96.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 7.3%

      \[\leadsto im \cdot \color{blue}{0.16666666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 15.9% accurate, 38.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 8.2 \cdot 10^{-29}:\\ \;\;\;\;im\_m \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot 0.027777777777777776\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= re 8.2e-29) (* im_m 0.0) (* im_m 0.027777777777777776))))

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 <= 8.2e-29) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.027777777777777776;
	}
	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 <= 8.2d-29) then
        tmp = im_m * 0.0d0
    else
        tmp = im_m * 0.027777777777777776d0
    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 <= 8.2e-29) {
		tmp = im_m * 0.0;
	} else {
		tmp = im_m * 0.027777777777777776;
	}
	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 <= 8.2e-29:
		tmp = im_m * 0.0
	else:
		tmp = im_m * 0.027777777777777776
	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 <= 8.2e-29)
		tmp = Float64(im_m * 0.0);
	else
		tmp = Float64(im_m * 0.027777777777777776);
	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 <= 8.2e-29)
		tmp = im_m * 0.0;
	else
		tmp = im_m * 0.027777777777777776;
	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, 8.2e-29], N[(im$95$m * 0.0), $MachinePrecision], N[(im$95$m * 0.027777777777777776), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.1999999999999996e-29

   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 91.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 91.5%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 91.5%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 91.5%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 91.5%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 91.5%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 93.6%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 17.3%

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

   if 8.1999999999999996e-29 < re

   1. Initial program 56.6%

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

   3. Taylor expanded in im around 0 95.8%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 95.8%

         \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

      2. associate-+r+ 95.8%

         \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

      3. *-commutative 95.8%

         \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

      4. +-commutative 95.8%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

      5. +-commutative 95.8%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

   5. Simplified 95.8%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

   7. Applied egg-rr 7.0%

      \[\leadsto im \cdot \color{blue}{0.027777777777777776} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 32.7% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in im around 0 47.1%

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

   1. associate-*r* 47.1%

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

   2. neg-mul-1 47.1%

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

5. Simplified 47.1%

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

6. Taylor expanded in re around 0 29.0%

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

   1. associate-*r* 29.0%

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

   2. neg-mul-1 29.0%

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

8. Simplified 29.0%

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

9. Final simplification 29.0%

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

Alternative 24: 15.3% accurate, 102.7× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m 0.0)))

C

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

Fortran

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

Java

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

Python

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

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 * 0.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in im around 0 92.7%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 92.7%

      \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

   2. associate-+r+ 92.7%

      \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

   3. *-commutative 92.7%

      \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

   4. +-commutative 92.7%

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. +-commutative 92.7%

      \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

5. Simplified 94.2%

   \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

7. Applied egg-rr 13.4%

   \[\leadsto im \cdot \color{blue}{0} \]
8. Add Preprocessing

Alternative 25: 5.7% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -2\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -2.0)))

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 * -2.0);
}

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 * (-2.0d0))
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 * -2.0);
}

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 * -2.0)

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 * -2.0))
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 * -2.0);
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 * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in im around 0 92.7%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 92.7%

      \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

   2. associate-+r+ 92.7%

      \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

   3. *-commutative 92.7%

      \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

   4. +-commutative 92.7%

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. +-commutative 92.7%

      \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

5. Simplified 94.2%

   \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

7. Applied egg-rr 5.1%

   \[\leadsto im \cdot \color{blue}{-2} \]
8. Add Preprocessing

Alternative 26: 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 #s(literal 1 binary64) 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 68.6%

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

3. Taylor expanded in im around 0 92.7%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 92.7%

      \[\leadsto im \cdot \left(-1 \cdot \sin re + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)}\right) \]

   2. associate-+r+ 92.7%

      \[\leadsto im \cdot \color{blue}{\left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot {im}^{2}\right)} \]

   3. *-commutative 92.7%

      \[\leadsto im \cdot \left(\left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)}\right) \]

   4. +-commutative 92.7%

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \left(-1 \cdot \sin re + \left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right)\right)} \]

   5. +-commutative 92.7%

      \[\leadsto im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2} + -1 \cdot \sin re\right)}\right) \]

5. Simplified 94.2%

   \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.0001984126984126984, -0.008333333333333333\right) \cdot {im}^{4} + \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

7. Applied egg-rr 5.0%

   \[\leadsto im \cdot \color{blue}{-3} \]
8. Add Preprocessing

Alternative 27: 5.6% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -4\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -4.0)))

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 * -4.0);
}

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 * (-4.0d0))
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 * -4.0);
}

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 * -4.0)

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 * -4.0))
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 * -4.0);
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 * -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in im around 0 47.1%

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

   1. associate-*r* 47.1%

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

   2. neg-mul-1 47.1%

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

5. Simplified 47.1%

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

7. Applied egg-rr 5.0%

   \[\leadsto \left(-im\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin re\right)} - -3\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 5.0%

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

   2. rem-exp-log 5.0%

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

   3. +-commutative 5.0%

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

   4. associate--l+ 5.0%

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

   5. metadata-eval 5.0%

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

9. Simplified 5.0%

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

10. Taylor expanded in re around 0 5.0%

    \[\leadsto \color{blue}{-4 \cdot im} \]
11. Step-by-step derivation

    1. *-commutative 5.0%

       \[\leadsto \color{blue}{im \cdot -4} \]

12. Simplified 5.0%

    \[\leadsto \color{blue}{im \cdot -4} \]
13. 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 2024107 
(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))))