math.sin on complex, imaginary part

Percentage Accurate: 55.6% → 99.5%

Time: 10.8s

Alternatives: 10

Speedup: 2.8×

• 51.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.008333333333333333\right) + -0.16666666666666666\right)\right)\right) - \cos re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -2e+105)
      (* (* 0.5 (cos re)) t_0)
      (*
       im_m
       (-
        (*
         im_m
         (*
          im_m
          (*
           (cos re)
           (+ (* im_m (* im_m -0.008333333333333333)) -0.16666666666666666))))
        (cos re)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -2e+105) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-2d+105)) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * (-0.008333333333333333d0))) + (-0.16666666666666666d0))))) - cos(re))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -2e+105) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = im_m * ((im_m * (im_m * (Math.cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - Math.cos(re));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -2e+105:
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = im_m * ((im_m * (im_m * (math.cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - math.cos(re))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -2e+105)
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(im_m * Float64(cos(re) * Float64(Float64(im_m * Float64(im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -2e+105)
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e+105], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -1.9999999999999999e105

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   if -1.9999999999999999e105 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 38.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 38.7%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. unpow2 95.5%

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

      3. associate-*r* 95.5%

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

      4. +-commutative 95.5%

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

      5. associate-*r* 95.5%

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

      6. distribute-rgt-out 95.5%

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

   7. Applied egg-rr 95.5%

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

      1. unpow2 95.5%

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

      2. associate-*r* 95.5%

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

   9. Applied egg-rr 95.5%

      \[\leadsto im \cdot \left(-1 \cdot \cos re + \left(\left(\cos re \cdot \left(\color{blue}{\left(-0.008333333333333333 \cdot im\right) \cdot im} + -0.16666666666666666\right)\right) \cdot im\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+105}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.008333333333333333\right) + -0.16666666666666666\right)\right)\right) - \cos re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left(-0.3333333333333333, {im\_m}^{2}, -2\right)\right)\right)\right)\right) \end{array} \]

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
  (*
   0.5
   (*
    im_m
    (log1p
     (expm1 (* (cos re) (fma -0.3333333333333333 (pow im_m 2.0) -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 * (0.5 * (im_m * log1p(expm1((cos(re) * fma(-0.3333333333333333, pow(im_m, 2.0), -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(0.5 * Float64(im_m * log1p(expm1(Float64(cos(re) * fma(-0.3333333333333333, (im_m ^ 2.0), -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[(0.5 * N[(im$95$m * N[Log[1 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(-0.3333333333333333 * N[Power[im$95$m, 2.0], $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 85.2%

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

6. Taylor expanded in re around inf 85.2%

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

   1. log1p-expm1-u 99.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)}\right) \]

   2. fma-neg 99.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)}\right)\right)\right) \]

   3. metadata-eval 99.4%

      \[\leadsto 0.5 \cdot \left(im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, \color{blue}{-2}\right)\right)\right)\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)\right)\right)}\right) \]
9. Add Preprocessing

Alternative 3: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 250:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.008333333333333333\right) + -0.16666666666666666\right)\right)\right) - \cos re\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

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 250.0)
    (*
     im_m
     (-
      (*
       im_m
       (*
        im_m
        (*
         (cos re)
         (+ (* im_m (* im_m -0.008333333333333333)) -0.16666666666666666))))
      (cos re)))
    (if (<= im_m 4.5e+61)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* (cos re) (* -0.008333333333333333 (pow im_m 5.0)))))))

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 <= 250.0) {
		tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = cos(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 250.0d0) then
        tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * (-0.008333333333333333d0))) + (-0.16666666666666666d0))))) - cos(re))
    else if (im_m <= 4.5d+61) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = cos(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 250.0) {
		tmp = im_m * ((im_m * (im_m * (Math.cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - Math.cos(re));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (-0.008333333333333333 * Math.pow(im_m, 5.0));
	}
	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 <= 250.0:
		tmp = im_m * ((im_m * (im_m * (math.cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - math.cos(re))
	elif im_m <= 4.5e+61:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = math.cos(re) * (-0.008333333333333333 * math.pow(im_m, 5.0))
	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 <= 250.0)
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(im_m * Float64(cos(re) * Float64(Float64(im_m * Float64(im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re)));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64(-0.008333333333333333 * (im_m ^ 5.0)));
	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 <= 250.0)
		tmp = im_m * ((im_m * (im_m * (cos(re) * ((im_m * (im_m * -0.008333333333333333)) + -0.16666666666666666)))) - cos(re));
	elseif (im_m <= 4.5e+61)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = cos(re) * (-0.008333333333333333 * (im_m ^ 5.0));
	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, 250.0], N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 250

   1. Initial program 39.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 39.1%

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

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 95.0%

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

      1. *-commutative 95.0%

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

      2. unpow2 95.0%

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

      3. associate-*r* 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-*r* 95.0%

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

      6. distribute-rgt-out 95.0%

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

   7. Applied egg-rr 95.0%

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

      1. unpow2 95.0%

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

      2. associate-*r* 95.0%

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

   9. Applied egg-rr 95.0%

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

   if 250 < im < 4.5e61

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 72.7%

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

   if 4.5e61 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.008333333333333333\right) + -0.16666666666666666\right)\right)\right) - \cos re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 250 \lor \neg \left(im\_m \leq 8.5 \cdot 10^{+102}\right):\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (or (<= im_m 250.0) (not (<= im_m 8.5e+102)))
    (* 0.5 (* im_m (* (cos re) (- (* im_m (* im_m -0.3333333333333333)) 2.0))))
    (* 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 <= 250.0) || !(im_m <= 8.5e+102)) {
		tmp = 0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	} else {
		tmp = 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 <= 250.0d0) .or. (.not. (im_m <= 8.5d+102))) then
        tmp = 0.5d0 * (im_m * (cos(re) * ((im_m * (im_m * (-0.3333333333333333d0))) - 2.0d0)))
    else
        tmp = 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 <= 250.0) || !(im_m <= 8.5e+102)) {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	} else {
		tmp = 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 <= 250.0) or not (im_m <= 8.5e+102):
		tmp = 0.5 * (im_m * (math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)))
	else:
		tmp = 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 <= 250.0) || !(im_m <= 8.5e+102))
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 2.0))));
	else
		tmp = Float64(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 <= 250.0) || ~((im_m <= 8.5e+102)))
		tmp = 0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	else
		tmp = 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[Or[LessEqual[im$95$m, 250.0], N[Not[LessEqual[im$95$m, 8.5e+102]], $MachinePrecision]], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * 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 < 250 or 8.4999999999999996e102 < im

   1. Initial program 49.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 49.0%

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

   3. Simplified 49.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 93.3%

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

   6. Taylor expanded in re around inf 93.3%

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

      1. unpow2 93.3%

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

      2. associate-*r* 93.3%

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

   8. Applied egg-rr 93.3%

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

   if 250 < im < 8.4999999999999996e102

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 83.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250 \lor \neg \left(im \leq 8.5 \cdot 10^{+102}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 250:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

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 250.0)
    (* 0.5 (* im_m (* (cos re) (- (* im_m (* im_m -0.3333333333333333)) 2.0))))
    (if (<= im_m 4.5e+61)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* (cos re) (* -0.008333333333333333 (pow im_m 5.0)))))))

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 <= 250.0) {
		tmp = 0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = cos(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	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 <= 250.0d0) then
        tmp = 0.5d0 * (im_m * (cos(re) * ((im_m * (im_m * (-0.3333333333333333d0))) - 2.0d0)))
    else if (im_m <= 4.5d+61) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = cos(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    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 <= 250.0) {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (-0.008333333333333333 * Math.pow(im_m, 5.0));
	}
	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 <= 250.0:
		tmp = 0.5 * (im_m * (math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)))
	elif im_m <= 4.5e+61:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = math.cos(re) * (-0.008333333333333333 * math.pow(im_m, 5.0))
	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 <= 250.0)
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 2.0))));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64(-0.008333333333333333 * (im_m ^ 5.0)));
	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 <= 250.0)
		tmp = 0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 2.0)));
	elseif (im_m <= 4.5e+61)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = cos(re) * (-0.008333333333333333 * (im_m ^ 5.0));
	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, 250.0], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 250

   1. Initial program 39.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 39.1%

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

   3. Simplified 39.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 92.0%

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

   6. Taylor expanded in re around inf 92.0%

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

      1. unpow2 92.0%

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

      2. associate-*r* 92.0%

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

   8. Applied egg-rr 92.0%

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

   if 250 < im < 4.5e61

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 72.7%

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

   if 4.5e61 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 250:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 \cdot im\_m\right)\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 (<= (cos re) -0.02)
    (- (* re (* re (* 0.5 im_m))) im_m)
    (* 0.5 (* im_m (- (* im_m (* im_m -0.3333333333333333)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = (re * (re * (0.5 * im_m))) - im_m;
	} else {
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (cos(re) <= (-0.02d0)) then
        tmp = (re * (re * (0.5d0 * im_m))) - im_m
    else
        tmp = 0.5d0 * (im_m * ((im_m * (im_m * (-0.3333333333333333d0))) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (Math.cos(re) <= -0.02) {
		tmp = (re * (re * (0.5 * im_m))) - im_m;
	} else {
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if math.cos(re) <= -0.02:
		tmp = (re * (re * (0.5 * im_m))) - im_m
	else:
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(re * Float64(re * Float64(0.5 * im_m))) - im_m);
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (cos(re) <= -0.02)
		tmp = (re * (re * (0.5 * im_m))) - im_m;
	else
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(re * N[(re * N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 51.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 51.5%

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

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 56.5%

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

      1. mul-1-neg 56.5%

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

      2. distribute-rgt-neg-in 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in re around 0 34.2%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left(im \cdot {re}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 34.2%

         \[\leadsto -1 \cdot im + \color{blue}{\left(0.5 \cdot im\right) \cdot {re}^{2}} \]

      2. unpow2 34.2%

         \[\leadsto -1 \cdot im + \left(0.5 \cdot im\right) \cdot \color{blue}{\left(re \cdot re\right)} \]

      3. associate-*r* 34.3%

         \[\leadsto -1 \cdot im + \color{blue}{\left(\left(0.5 \cdot im\right) \cdot re\right) \cdot re} \]

      4. *-commutative 34.3%

         \[\leadsto -1 \cdot im + \left(\color{blue}{\left(im \cdot 0.5\right)} \cdot re\right) \cdot re \]

   10. Applied egg-rr 34.3%

       \[\leadsto -1 \cdot im + \color{blue}{\left(\left(im \cdot 0.5\right) \cdot re\right) \cdot re} \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 54.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 54.7%

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

   3. Simplified 54.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 82.7%

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

   6. Taylor expanded in re around 0 68.9%

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

      1. unpow2 82.7%

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

      2. associate-*r* 82.7%

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

   8. Applied egg-rr 68.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;re \cdot \left(re \cdot \left(0.5 \cdot im\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\right)\right) \end{array} \]

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
  (* 0.5 (* im_m (* (cos re) (- (* im_m (* im_m -0.3333333333333333)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 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 * (0.5d0 * (im_m * (cos(re) * ((im_m * (im_m * (-0.3333333333333333d0))) - 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 * (0.5 * (im_m * (Math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 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 * (0.5 * (im_m * (math.cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 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(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 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 * (0.5 * (im_m * (cos(re) * ((im_m * (im_m * -0.3333333333333333)) - 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[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 85.2%

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

6. Taylor expanded in re around inf 85.2%

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

   1. unpow2 85.2%

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

   2. associate-*r* 85.2%

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

8. Applied egg-rr 85.2%

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

9. Final simplification 85.2%

   \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.7 \cdot 10^{+36}:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \]

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 2.7e+36)
    (* (cos re) (- im_m))
    (* 0.5 (* im_m (- (* im_m (* im_m -0.3333333333333333)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.7e+36) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.7d+36) then
        tmp = cos(re) * -im_m
    else
        tmp = 0.5d0 * (im_m * ((im_m * (im_m * (-0.3333333333333333d0))) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.7e+36) {
		tmp = Math.cos(re) * -im_m;
	} else {
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.7e+36:
		tmp = math.cos(re) * -im_m
	else:
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.7e+36)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.7e+36)
		tmp = cos(re) * -im_m;
	else
		tmp = 0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.7e+36], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.7000000000000001e36

   1. Initial program 41.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 41.8%

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

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. distribute-rgt-neg-in 65.0%

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

   7. Simplified 65.0%

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

   if 2.7000000000000001e36 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 73.8%

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

   6. Taylor expanded in re around 0 53.0%

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

      1. unpow2 73.8%

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

      2. associate-*r* 73.8%

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

   8. Applied egg-rr 53.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7 \cdot 10^{+36}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right) - 2\right)\right)\right) \end{array} \]

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 (* 0.5 (* im_m (- (* im_m (* im_m -0.3333333333333333)) 2.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 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 * (0.5d0 * (im_m * ((im_m * (im_m * (-0.3333333333333333d0))) - 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 * (0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 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 * (0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 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(0.5 * Float64(im_m * Float64(Float64(im_m * Float64(im_m * -0.3333333333333333)) - 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 * (0.5 * (im_m * ((im_m * (im_m * -0.3333333333333333)) - 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[(0.5 * N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 85.2%

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

6. Taylor expanded in re around 0 50.4%

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

   1. unpow2 85.2%

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

   2. associate-*r* 85.2%

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

8. Applied egg-rr 50.4%

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

9. Final simplification 50.4%

   \[\leadsto 0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.3333333333333333\right) - 2\right)\right) \]
10. Add Preprocessing

Alternative 10: 29.2% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. neg-sub0 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.7%

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

   1. mul-1-neg 52.7%

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

   2. distribute-rgt-neg-in 52.7%

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

7. Simplified 52.7%

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

8. Taylor expanded in re around 0 27.7%

   \[\leadsto \color{blue}{-1 \cdot im} \]
9. Step-by-step derivation

   1. mul-1-neg 27.7%

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

10. Simplified 27.7%

    \[\leadsto \color{blue}{-im} \]
11. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - 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.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(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 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[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[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))