math.sin on complex, imaginary part

Percentage Accurate: 53.8% → 99.7%

Time: 8.2s

Alternatives: 11

Speedup: 2.9×

• 49.1% 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: 53.8% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;0.5 \cdot \left(t_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 + {im_m}^{5} \cdot -0.008333333333333333\right) - im_m \cdot \cos re\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -100.0)
      (* 0.5 (* t_0 (cos re)))
      (-
       (*
        (cos re)
        (+
         (* (pow im_m 3.0) -0.16666666666666666)
         (* (pow im_m 5.0) -0.008333333333333333)))
       (* im_m (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 <= -100.0) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = (cos(re) * ((pow(im_m, 3.0) * -0.16666666666666666) + (pow(im_m, 5.0) * -0.008333333333333333))) - (im_m * 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 <= (-100.0d0)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = (cos(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) + ((im_m ** 5.0d0) * (-0.008333333333333333d0)))) - (im_m * 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 <= -100.0) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = (Math.cos(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) + (Math.pow(im_m, 5.0) * -0.008333333333333333))) - (im_m * 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 <= -100.0:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = (math.cos(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) + (math.pow(im_m, 5.0) * -0.008333333333333333))) - (im_m * 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 <= -100.0)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(Float64(cos(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) + Float64((im_m ^ 5.0) * -0.008333333333333333))) - Float64(im_m * 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 <= -100.0)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = (cos(re) * (((im_m ^ 3.0) * -0.16666666666666666) + ((im_m ^ 5.0) * -0.008333333333333333))) - (im_m * 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, -100.0], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -100

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   if -100 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 96.1%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.1%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 96.1%

         \[\leadsto \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 96.1%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) - im \cdot \cos re} \]

      4. associate-*r* 96.1%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) - im \cdot \cos re \]

      5. associate-*r* 96.1%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) - im \cdot \cos re \]

      6. distribute-rgt-out 96.1%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + -0.008333333333333333 \cdot {im}^{5}\right)} - im \cdot \cos re \]

      7. *-commutative 96.1%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} + -0.008333333333333333 \cdot {im}^{5}\right) - im \cdot \cos re \]

      8. *-commutative 96.1%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \color{blue}{{im}^{5} \cdot -0.008333333333333333}\right) - im \cdot \cos re \]

   6. Simplified 96.1%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \cos re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -100:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \cos re\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -100:\\ \;\;\;\;0.5 \cdot \left(t_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im_m \cdot -2 + \left({im_m}^{3} \cdot -0.3333333333333333 + {im_m}^{5} \cdot -0.016666666666666666\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -100.0)
      (* 0.5 (* t_0 (cos re)))
      (*
       0.5
       (*
        (cos re)
        (+
         (* im_m -2.0)
         (+
          (* (pow im_m 3.0) -0.3333333333333333)
          (* (pow im_m 5.0) -0.016666666666666666)))))))))

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 <= -100.0) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = 0.5 * (cos(re) * ((im_m * -2.0) + ((pow(im_m, 3.0) * -0.3333333333333333) + (pow(im_m, 5.0) * -0.016666666666666666))));
	}
	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 <= (-100.0d0)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = 0.5d0 * (cos(re) * ((im_m * (-2.0d0)) + (((im_m ** 3.0d0) * (-0.3333333333333333d0)) + ((im_m ** 5.0d0) * (-0.016666666666666666d0)))))
    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 <= -100.0) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = 0.5 * (Math.cos(re) * ((im_m * -2.0) + ((Math.pow(im_m, 3.0) * -0.3333333333333333) + (Math.pow(im_m, 5.0) * -0.016666666666666666))));
	}
	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 <= -100.0:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = 0.5 * (math.cos(re) * ((im_m * -2.0) + ((math.pow(im_m, 3.0) * -0.3333333333333333) + (math.pow(im_m, 5.0) * -0.016666666666666666))))
	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 <= -100.0)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im_m * -2.0) + Float64(Float64((im_m ^ 3.0) * -0.3333333333333333) + Float64((im_m ^ 5.0) * -0.016666666666666666)))));
	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 <= -100.0)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = 0.5 * (cos(re) * ((im_m * -2.0) + (((im_m ^ 3.0) * -0.3333333333333333) + ((im_m ^ 5.0) * -0.016666666666666666))));
	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, -100.0], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * -2.0), $MachinePrecision] + N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -100

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   if -100 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 96.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \cdot \cos re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -100:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(t_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.01)
      (* 0.5 (* t_0 (cos re)))
      (* (cos re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = cos(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.01d0)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = cos(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = Math.cos(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.01:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = math.cos(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(cos(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = cos(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.01], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0100000000000000002

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   if -0.0100000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 91.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. associate-*r* 91.0%

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

      5. distribute-rgt-out-- 91.0%

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

      6. *-commutative 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.006:\\ \;\;\;\;\left(-im_m\right) \cdot \cos re\\ \mathbf{elif}\;im_m \leq 3.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(e^{-im_m} - e^{im_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im_m}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.006)
    (* (- im_m) (cos re))
    (if (<= im_m 3.2e+42)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* 0.5 (* -0.0003968253968253968 (* (cos 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 <= 0.006) {
		tmp = -im_m * cos(re);
	} else if (im_m <= 3.2e+42) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(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 <= 0.006d0) then
        tmp = -im_m * cos(re)
    else if (im_m <= 3.2d+42) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (cos(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 <= 0.006) {
		tmp = -im_m * Math.cos(re);
	} else if (im_m <= 3.2e+42) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(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 <= 0.006:
		tmp = -im_m * math.cos(re)
	elif im_m <= 3.2e+42:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(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 <= 0.006)
		tmp = Float64(Float64(-im_m) * cos(re));
	elseif (im_m <= 3.2e+42)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(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 <= 0.006)
		tmp = -im_m * cos(re);
	elseif (im_m <= 3.2e+42)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (cos(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, 0.006], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.2e+42], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0060000000000000001

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

   if 0.0060000000000000001 < im < 3.20000000000000002e42

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 68.4%

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

      1. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if 3.20000000000000002e42 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.006:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]

Alternative 5: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.0095:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666 - im_m\right)\\ \mathbf{elif}\;im_m \leq 3.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(e^{-im_m} - e^{im_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im_m}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.0095)
    (* (cos re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 3.2e+42)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* 0.5 (* -0.0003968253968253968 (* (cos 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 <= 0.0095) {
		tmp = cos(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 3.2e+42) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(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 <= 0.0095d0) then
        tmp = cos(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 3.2d+42) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (cos(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 <= 0.0095) {
		tmp = Math.cos(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 3.2e+42) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(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 <= 0.0095:
		tmp = math.cos(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 3.2e+42:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(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 <= 0.0095)
		tmp = Float64(cos(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 3.2e+42)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(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 <= 0.0095)
		tmp = cos(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 3.2e+42)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (cos(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, 0.0095], N[(N[Cos[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, 3.2e+42], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.00949999999999999976

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 91.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. associate-*r* 91.0%

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

      5. distribute-rgt-out-- 91.0%

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

      6. *-commutative 91.0%

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

   6. Simplified 91.0%

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

   if 0.00949999999999999976 < im < 3.20000000000000002e42

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 68.4%

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

      1. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if 3.20000000000000002e42 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{+42}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]

Alternative 6: 90.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;\left(-im_m\right) \cdot \cos re\\ \mathbf{elif}\;im_m \leq 1.04 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im_m}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.6e+20)
    (* (- im_m) (cos re))
    (if (<= im_m 1.04e+102)
      (* 0.5 (* -0.0003968253968253968 (pow im_m 7.0)))
      (* (cos re) (* (pow im_m 3.0) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6e+20) {
		tmp = -im_m * cos(re);
	} else if (im_m <= 1.04e+102) {
		tmp = 0.5 * (-0.0003968253968253968 * pow(im_m, 7.0));
	} else {
		tmp = cos(re) * (pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.6d+20) then
        tmp = -im_m * cos(re)
    else if (im_m <= 1.04d+102) then
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (im_m ** 7.0d0))
    else
        tmp = cos(re) * ((im_m ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6e+20) {
		tmp = -im_m * Math.cos(re);
	} else if (im_m <= 1.04e+102) {
		tmp = 0.5 * (-0.0003968253968253968 * Math.pow(im_m, 7.0));
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.6e+20:
		tmp = -im_m * math.cos(re)
	elif im_m <= 1.04e+102:
		tmp = 0.5 * (-0.0003968253968253968 * math.pow(im_m, 7.0))
	else:
		tmp = math.cos(re) * (math.pow(im_m, 3.0) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.6e+20)
		tmp = Float64(Float64(-im_m) * cos(re));
	elseif (im_m <= 1.04e+102)
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (im_m ^ 7.0)));
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 3.0) * -0.16666666666666666));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.6e+20)
		tmp = -im_m * cos(re);
	elseif (im_m <= 1.04e+102)
		tmp = 0.5 * (-0.0003968253968253968 * (im_m ^ 7.0));
	else
		tmp = cos(re) * ((im_m ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.6e+20], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.04e+102], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.6e20

   1. Initial program 40.4%

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

      1. sub-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 40.4%

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

      3. remove-double-neg 40.4%

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

      4. remove-double-neg 40.4%

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

      5. sub0-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 40.4%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 40.4%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 40.4%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 40.4%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 40.4%

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

   4. Taylor expanded in im around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

   6. Simplified 66.7%

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

   if 4.6e20 < im < 1.04e102

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   5. Taylor expanded in im around inf 78.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   6. Taylor expanded in re around 0 71.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]

   if 1.04e102 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 97.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. associate-*r* 97.7%

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

      5. distribute-rgt-out-- 97.7%

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

      6. *-commutative 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in im around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. *-commutative 97.7%

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

      3. associate-*r* 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 1.04 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 7: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.0062:\\ \;\;\;\;\left(-im_m\right) \cdot \cos re\\ \mathbf{elif}\;im_m \leq 1.04 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im_m} - e^{im_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.0062)
    (* (- im_m) (cos re))
    (if (<= im_m 1.04e+102)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* (cos re) (* (pow im_m 3.0) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0062) {
		tmp = -im_m * cos(re);
	} else if (im_m <= 1.04e+102) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = cos(re) * (pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.0062d0) then
        tmp = -im_m * cos(re)
    else if (im_m <= 1.04d+102) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = cos(re) * ((im_m ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0062) {
		tmp = -im_m * Math.cos(re);
	} else if (im_m <= 1.04e+102) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.0062:
		tmp = -im_m * math.cos(re)
	elif im_m <= 1.04e+102:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = math.cos(re) * (math.pow(im_m, 3.0) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.0062)
		tmp = Float64(Float64(-im_m) * cos(re));
	elseif (im_m <= 1.04e+102)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 3.0) * -0.16666666666666666));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.0062)
		tmp = -im_m * cos(re);
	elseif (im_m <= 1.04e+102)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = cos(re) * ((im_m ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.0062], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.04e+102], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.00619999999999999978

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 38.9%

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

      3. remove-double-neg 38.9%

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

      4. remove-double-neg 38.9%

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

      5. sub0-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 38.9%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 38.9%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 38.9%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 38.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 38.9%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 38.9%

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

   4. Taylor expanded in im around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. neg-mul-1 68.2%

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

   6. Simplified 68.2%

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

   if 0.00619999999999999978 < im < 1.04e102

   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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.5%

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

      1. *-commutative 82.5%

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

   6. Simplified 82.5%

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

   if 1.04e102 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 97.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. associate-*r* 97.7%

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

      5. distribute-rgt-out-- 97.7%

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

      6. *-commutative 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in im around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. *-commutative 97.7%

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

      3. associate-*r* 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0062:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 1.04 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 8: 82.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(-im_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5e+20)
    (* (- im_m) (cos re))
    (* 0.5 (* -0.0003968253968253968 (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 <= 5e+20) {
		tmp = -im_m * cos(re);
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * 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 <= 5d+20) then
        tmp = -im_m * cos(re)
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (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 <= 5e+20) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * 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 <= 5e+20:
		tmp = -im_m * math.cos(re)
	else:
		tmp = 0.5 * (-0.0003968253968253968 * 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 <= 5e+20)
		tmp = Float64(Float64(-im_m) * cos(re));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * (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 <= 5e+20)
		tmp = -im_m * cos(re);
	else
		tmp = 0.5 * (-0.0003968253968253968 * (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, 5e+20], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5e20

   1. Initial program 40.4%

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

      1. sub-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 40.4%

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

      3. remove-double-neg 40.4%

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

      4. remove-double-neg 40.4%

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

      5. sub0-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 40.4%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 40.4%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 40.4%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 40.4%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 40.4%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 40.4%

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

   4. Taylor expanded in im around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

   6. Simplified 66.7%

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

   if 5e20 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 93.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   5. Taylor expanded in im around inf 93.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   6. Taylor expanded in re around 0 76.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 9: 54.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 4.8:\\ \;\;\;\;-im_m\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 4.8) (- im_m) (* (pow im_m 3.0) -0.16666666666666666))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = -im_m;
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.8d0) then
        tmp = -im_m
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = -im_m;
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.8:
		tmp = -im_m
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.8)
		tmp = Float64(-im_m);
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.8)
		tmp = -im_m;
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.8], (-im$95$m), N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999982

   1. Initial program 39.2%

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

      1. sub-neg 39.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 39.2%

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

      3. remove-double-neg 39.2%

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

      4. remove-double-neg 39.2%

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

      5. sub0-neg 39.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 39.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 39.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 39.2%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 39.2%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 39.2%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 39.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 39.2%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 39.2%

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

   4. Taylor expanded in im around 0 68.0%

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

      1. associate-*r* 68.0%

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

      2. neg-mul-1 68.0%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in re around 0 45.4%

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

      1. mul-1-neg 45.4%

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

   9. Simplified 45.4%

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

   if 4.79999999999999982 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 65.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 65.0%

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

      3. unsub-neg 65.0%

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

      4. associate-*r* 65.0%

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

      5. distribute-rgt-out-- 65.0%

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

      6. *-commutative 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in im around inf 65.0%

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

      1. *-commutative 65.0%

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

      2. *-commutative 65.0%

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

      3. associate-*r* 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in re around 0 52.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 10: 75.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\left(-im_m\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.25e+54)
    (* (- im_m) (cos re))
    (* (pow im_m 3.0) -0.16666666666666666))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.25e+54) {
		tmp = -im_m * cos(re);
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.25d+54) then
        tmp = -im_m * cos(re)
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.25e+54) {
		tmp = -im_m * Math.cos(re);
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.25e+54:
		tmp = -im_m * math.cos(re)
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.25e+54)
		tmp = Float64(Float64(-im_m) * cos(re));
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.25e+54)
		tmp = -im_m * cos(re);
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.25e+54], N[((-im$95$m) * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.25000000000000001e54

   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. sub-neg 41.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 41.8%

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

      3. remove-double-neg 41.8%

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

      4. remove-double-neg 41.8%

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

      5. sub0-neg 41.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 41.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 41.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 41.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 41.8%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 41.8%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 41.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 41.8%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 41.8%

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

   4. Taylor expanded in im around 0 65.1%

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

      1. associate-*r* 65.1%

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

      2. neg-mul-1 65.1%

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

   6. Simplified 65.1%

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

   if 1.25000000000000001e54 < 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. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

      6. distribute-neg-in 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

      7. +-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

      8. sub-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

      9. cos-neg 100.0%

         \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

      11. distribute-rgt-neg-in 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

      12. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 76.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. associate-*r* 76.0%

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

      5. distribute-rgt-out-- 76.0%

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

      6. *-commutative 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in im around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in re around 0 61.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 11: 30.2% 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 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (- im_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. sub-neg 53.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)} \]

   2. neg-sub0 53.2%

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

   3. remove-double-neg 53.2%

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

   4. remove-double-neg 53.2%

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

   5. sub0-neg 53.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\left(-\left(-e^{-im}\right)\right) + \left(-e^{\color{blue}{0 - \left(-im\right)}}\right)\right) \]

   6. distribute-neg-in 53.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\left(-e^{-im}\right) + e^{0 - \left(-im\right)}\right)\right)} \]

   7. +-commutative 53.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} + \left(-e^{-im}\right)\right)}\right) \]

   8. sub-neg 53.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right)}\right) \]

   9. cos-neg 53.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right) \]

   10. associate-*l* 53.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(-\left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)\right)} \]

   11. distribute-rgt-neg-in 53.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-\cos \left(-re\right) \cdot \left(e^{0 - \left(-im\right)} - e^{-im}\right)\right)} \]

   12. *-commutative 53.2%

       \[\leadsto 0.5 \cdot \left(-\color{blue}{\left(e^{0 - \left(-im\right)} - e^{-im}\right) \cdot \cos \left(-re\right)}\right) \]

3. Simplified 53.2%

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

4. Taylor expanded in im around 0 53.6%

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

   1. associate-*r* 53.6%

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

   2. neg-mul-1 53.6%

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

6. Simplified 53.6%

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

7. Taylor expanded in re around 0 36.0%

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

   1. mul-1-neg 36.0%

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

9. Simplified 36.0%

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

10. Final simplification 36.0%

    \[\leadsto -im \]

Developer target: 99.8% accurate, 0.8× 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 2023333 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (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))))