math.sin on complex, imaginary part

Percentage Accurate: 53.0% → 99.7%

Time: 10.3s

Alternatives: 12

Speedup: 2.8×

• 48.3% 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: 53.0% 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, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \log \left(e^{\cos re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -10.0)
      (* 0.5 (* t_0 (log (exp (cos re)))))
      (*
       0.5
       (*
        (cos re)
        (*
         im_m
         (-
          (*
           (* im_m im_m)
           (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
          2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 0.5 * (t_0 * log(exp(cos(re))));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-10.0d0)) then
        tmp = 0.5d0 * (t_0 * log(exp(cos(re))))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 0.5 * (t_0 * Math.log(Math.exp(Math.cos(re))));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -10.0:
		tmp = 0.5 * (t_0 * math.log(math.exp(math.cos(re))))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(0.5 * Float64(t_0 * log(exp(cos(re)))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = 0.5 * (t_0 * log(exp(cos(re))));
	else
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -10.0], N[(0.5 * N[(t$95$0 * N[Log[N[Exp[N[Cos[re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -10

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -10 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 44.4%

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

      1. /-rgt-identity 44.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.4%

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

      3. associate-*l/ 44.4%

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

      4. cos-neg 44.4%

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

      5. associate-*l* 44.4%

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

      6. associate-*r/ 44.4%

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

      7. exp-0 44.4%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.4%

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

      9. *-commutative 44.4%

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

      10. neg-sub0 44.4%

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

      11. cos-neg 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 91.6%

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

      1. unpow2 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. unpow2 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \log \left(e^{\cos re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -10.0)
      (* 0.5 (* t_0 (cos re)))
      (*
       0.5
       (*
        (cos re)
        (*
         im_m
         (-
          (*
           (* im_m im_m)
           (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
          2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-10.0d0)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -10.0:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -10.0], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -10

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   if -10 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 44.4%

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

      1. /-rgt-identity 44.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.4%

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

      3. associate-*l/ 44.4%

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

      4. cos-neg 44.4%

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

      5. associate-*l* 44.4%

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

      6. associate-*r/ 44.4%

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

      7. exp-0 44.4%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.4%

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

      9. *-commutative 44.4%

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

      10. neg-sub0 44.4%

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

      11. cos-neg 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 91.6%

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

      1. unpow2 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. unpow2 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(0.3333333333333333 - e^{im\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.5)
    (*
     0.5
     (*
      (cos re)
      (*
       im_m
       (-
        (*
         (* im_m im_m)
         (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
        2.0))))
    (* 0.5 (* (cos re) (- 0.3333333333333333 (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5) {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (cos(re) * (0.3333333333333333 - exp(im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.5d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    else
        tmp = 0.5d0 * (cos(re) * (0.3333333333333333d0 - exp(im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5) {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (0.3333333333333333 - Math.exp(im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.5:
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))
	else:
		tmp = 0.5 * (math.cos(re) * (0.3333333333333333 - math.exp(im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.5)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(0.3333333333333333 - exp(im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.5)
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	else
		tmp = 0.5 * (cos(re) * (0.3333333333333333 - exp(im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.5], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(0.3333333333333333 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.5

   1. Initial program 44.4%

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

      1. /-rgt-identity 44.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.4%

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

      3. associate-*l/ 44.4%

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

      4. cos-neg 44.4%

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

      5. associate-*l* 44.4%

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

      6. associate-*r/ 44.4%

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

      7. exp-0 44.4%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.4%

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

      9. *-commutative 44.4%

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

      10. neg-sub0 44.4%

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

      11. cos-neg 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in im around 0 91.6%

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

      1. unpow2 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. unpow2 91.6%

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

   9. Applied egg-rr 91.6%

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

   if 2.5 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(0.3333333333333333 - e^{im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 600 \lor \neg \left(im\_m \leq 2.4 \cdot 10^{+96}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\left(5 + -0.5 \cdot {re}^{2}\right) \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 600.0) (not (<= im_m 2.4e+96)))
    (*
     0.5
     (*
      (cos re)
      (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0))))
    (* im_m (* (+ 5.0 (* -0.5 (pow re 2.0))) (+ -1.0 (* im_m -0.25)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 600.0) || !(im_m <= 2.4e+96)) {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	} else {
		tmp = im_m * ((5.0 + (-0.5 * pow(re, 2.0))) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 600.0d0) .or. (.not. (im_m <= 2.4d+96))) then
        tmp = 0.5d0 * (cos(re) * (im_m * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 2.0d0)))
    else
        tmp = im_m * ((5.0d0 + ((-0.5d0) * (re ** 2.0d0))) * ((-1.0d0) + (im_m * (-0.25d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 600.0) || !(im_m <= 2.4e+96)) {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	} else {
		tmp = im_m * ((5.0 + (-0.5 * Math.pow(re, 2.0))) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 600.0) or not (im_m <= 2.4e+96):
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)))
	else:
		tmp = im_m * ((5.0 + (-0.5 * math.pow(re, 2.0))) * (-1.0 + (im_m * -0.25)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 600.0) || !(im_m <= 2.4e+96))
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0))));
	else
		tmp = Float64(im_m * Float64(Float64(5.0 + Float64(-0.5 * (re ^ 2.0))) * Float64(-1.0 + Float64(im_m * -0.25))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 600.0) || ~((im_m <= 2.4e+96)))
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	else
		tmp = im_m * ((5.0 + (-0.5 * (re ^ 2.0))) * (-1.0 + (im_m * -0.25)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 600.0], N[Not[LessEqual[im$95$m, 2.4e+96]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(5.0 + N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 600 or 2.39999999999999993e96 < im

   1. Initial program 53.1%

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

      1. /-rgt-identity 53.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 53.1%

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

      3. associate-*l/ 53.1%

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

      4. cos-neg 53.1%

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

      5. associate-*l* 53.1%

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

      6. associate-*r/ 53.1%

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

      7. exp-0 53.1%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 53.1%

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

      9. *-commutative 53.1%

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

      10. neg-sub0 53.1%

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

      11. cos-neg 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in im around 0 21.5%

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

      1. neg-mul-1 21.5%

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

      2. unsub-neg 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in im around 0 86.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \cdot \cos re\right) \]

   if 600 < im < 2.39999999999999993e96

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 4.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 4.1%

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

      2. distribute-rgt-out 4.1%

         \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   10. Simplified 4.1%

       \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   12. Applied egg-rr 2.6%

       \[\leadsto im \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos re\right)} - -3\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]
   13. Step-by-step derivation

       1. sub-neg 2.6%

          \[\leadsto im \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos re\right)} + \left(--3\right)\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       2. log1p-undefine 2.6%

          \[\leadsto im \cdot \left(\left(e^{\color{blue}{\log \left(1 + \cos re\right)}} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       3. rem-exp-log 2.6%

          \[\leadsto im \cdot \left(\left(\color{blue}{\left(1 + \cos re\right)} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       4. +-commutative 2.6%

          \[\leadsto im \cdot \left(\left(\color{blue}{\left(\cos re + 1\right)} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       5. metadata-eval 2.6%

          \[\leadsto im \cdot \left(\left(\left(\cos re + 1\right) + \color{blue}{3}\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   14. Simplified 2.6%

       \[\leadsto im \cdot \left(\color{blue}{\left(\left(\cos re + 1\right) + 3\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   15. Taylor expanded in re around 0 31.2%

       \[\leadsto im \cdot \left(\color{blue}{\left(5 + -0.5 \cdot {re}^{2}\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600 \lor \neg \left(im \leq 2.4 \cdot 10^{+96}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(5 + -0.5 \cdot {re}^{2}\right) \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -1 + im\_m \cdot -0.25\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 980:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot t\_0\right)\\ \mathbf{elif}\;im\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;im\_m \cdot \left(\left(5 + -0.5 \cdot {re}^{2}\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* im_m -0.25))))
   (*
    im_s
    (if (<= im_m 980.0)
      (* im_m (* (cos re) t_0))
      (if (<= im_m 4e+150)
        (* im_m (* (+ 5.0 (* -0.5 (pow re 2.0))) t_0))
        (* 0.5 (* (cos re) (* im_m (- (* im_m -0.5) 2.0)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -1.0 + (im_m * -0.25);
	double tmp;
	if (im_m <= 980.0) {
		tmp = im_m * (cos(re) * t_0);
	} else if (im_m <= 4e+150) {
		tmp = im_m * ((5.0 + (-0.5 * pow(re, 2.0))) * t_0);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (im_m * (-0.25d0))
    if (im_m <= 980.0d0) then
        tmp = im_m * (cos(re) * t_0)
    else if (im_m <= 4d+150) then
        tmp = im_m * ((5.0d0 + ((-0.5d0) * (re ** 2.0d0))) * t_0)
    else
        tmp = 0.5d0 * (cos(re) * (im_m * ((im_m * (-0.5d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -1.0 + (im_m * -0.25);
	double tmp;
	if (im_m <= 980.0) {
		tmp = im_m * (Math.cos(re) * t_0);
	} else if (im_m <= 4e+150) {
		tmp = im_m * ((5.0 + (-0.5 * Math.pow(re, 2.0))) * t_0);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -1.0 + (im_m * -0.25)
	tmp = 0
	if im_m <= 980.0:
		tmp = im_m * (math.cos(re) * t_0)
	elif im_m <= 4e+150:
		tmp = im_m * ((5.0 + (-0.5 * math.pow(re, 2.0))) * t_0)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-1.0 + Float64(im_m * -0.25))
	tmp = 0.0
	if (im_m <= 980.0)
		tmp = Float64(im_m * Float64(cos(re) * t_0));
	elseif (im_m <= 4e+150)
		tmp = Float64(im_m * Float64(Float64(5.0 + Float64(-0.5 * (re ^ 2.0))) * t_0));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * -0.5) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = -1.0 + (im_m * -0.25);
	tmp = 0.0;
	if (im_m <= 980.0)
		tmp = im_m * (cos(re) * t_0);
	elseif (im_m <= 4e+150)
		tmp = im_m * ((5.0 + (-0.5 * (re ^ 2.0))) * t_0);
	else
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 980.0], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4e+150], N[(im$95$m * N[(N[(5.0 + N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 980

   1. Initial program 44.7%

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

      1. /-rgt-identity 44.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.7%

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

      3. associate-*l/ 44.7%

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

      4. cos-neg 44.7%

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

      5. associate-*l* 44.7%

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

      6. associate-*r/ 44.7%

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

      7. exp-0 44.7%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.7%

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

      9. *-commutative 44.7%

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

      10. neg-sub0 44.7%

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

      11. cos-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in im around 0 7.3%

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

      1. neg-mul-1 7.3%

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

      2. unsub-neg 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in im around 0 59.7%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 59.7%

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

      2. distribute-rgt-out 59.7%

         \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   10. Simplified 59.7%

       \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   if 980 < im < 3.99999999999999992e150

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 5.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 5.1%

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

      2. distribute-rgt-out 5.1%

         \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   10. Simplified 5.1%

       \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   12. Applied egg-rr 3.4%

       \[\leadsto im \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos re\right)} - -3\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]
   13. Step-by-step derivation

       1. sub-neg 3.4%

          \[\leadsto im \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos re\right)} + \left(--3\right)\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       2. log1p-undefine 3.4%

          \[\leadsto im \cdot \left(\left(e^{\color{blue}{\log \left(1 + \cos re\right)}} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       3. rem-exp-log 3.4%

          \[\leadsto im \cdot \left(\left(\color{blue}{\left(1 + \cos re\right)} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       4. +-commutative 3.4%

          \[\leadsto im \cdot \left(\left(\color{blue}{\left(\cos re + 1\right)} + \left(--3\right)\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

       5. metadata-eval 3.4%

          \[\leadsto im \cdot \left(\left(\left(\cos re + 1\right) + \color{blue}{3}\right) \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   14. Simplified 3.4%

       \[\leadsto im \cdot \left(\color{blue}{\left(\left(\cos re + 1\right) + 3\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   15. Taylor expanded in re around 0 31.7%

       \[\leadsto im \cdot \left(\color{blue}{\left(5 + -0.5 \cdot {re}^{2}\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   if 3.99999999999999992e150 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 97.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(\left(5 + -0.5 \cdot {re}^{2}\right) \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -1 + im\_m \cdot -0.25\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 410:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot t\_0\right)\\ \mathbf{elif}\;im\_m \leq 4 \cdot 10^{+150}:\\ \;\;\;\;im\_m \cdot \left(t\_0 \cdot \left(-0.5 \cdot {re}^{2} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* im_m -0.25))))
   (*
    im_s
    (if (<= im_m 410.0)
      (* im_m (* (cos re) t_0))
      (if (<= im_m 4e+150)
        (* im_m (* t_0 (+ (* -0.5 (pow re 2.0)) 1.0)))
        (* 0.5 (* (cos re) (* im_m (- (* im_m -0.5) 2.0)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -1.0 + (im_m * -0.25);
	double tmp;
	if (im_m <= 410.0) {
		tmp = im_m * (cos(re) * t_0);
	} else if (im_m <= 4e+150) {
		tmp = im_m * (t_0 * ((-0.5 * pow(re, 2.0)) + 1.0));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (im_m * (-0.25d0))
    if (im_m <= 410.0d0) then
        tmp = im_m * (cos(re) * t_0)
    else if (im_m <= 4d+150) then
        tmp = im_m * (t_0 * (((-0.5d0) * (re ** 2.0d0)) + 1.0d0))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * ((im_m * (-0.5d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = -1.0 + (im_m * -0.25);
	double tmp;
	if (im_m <= 410.0) {
		tmp = im_m * (Math.cos(re) * t_0);
	} else if (im_m <= 4e+150) {
		tmp = im_m * (t_0 * ((-0.5 * Math.pow(re, 2.0)) + 1.0));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = -1.0 + (im_m * -0.25)
	tmp = 0
	if im_m <= 410.0:
		tmp = im_m * (math.cos(re) * t_0)
	elif im_m <= 4e+150:
		tmp = im_m * (t_0 * ((-0.5 * math.pow(re, 2.0)) + 1.0))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(-1.0 + Float64(im_m * -0.25))
	tmp = 0.0
	if (im_m <= 410.0)
		tmp = Float64(im_m * Float64(cos(re) * t_0));
	elseif (im_m <= 4e+150)
		tmp = Float64(im_m * Float64(t_0 * Float64(Float64(-0.5 * (re ^ 2.0)) + 1.0)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * -0.5) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = -1.0 + (im_m * -0.25);
	tmp = 0.0;
	if (im_m <= 410.0)
		tmp = im_m * (cos(re) * t_0);
	elseif (im_m <= 4e+150)
		tmp = im_m * (t_0 * ((-0.5 * (re ^ 2.0)) + 1.0));
	else
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 410.0], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4e+150], N[(im$95$m * N[(t$95$0 * N[(N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 410

   1. Initial program 44.7%

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

      1. /-rgt-identity 44.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 44.7%

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

      3. associate-*l/ 44.7%

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

      4. cos-neg 44.7%

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

      5. associate-*l* 44.7%

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

      6. associate-*r/ 44.7%

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

      7. exp-0 44.7%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 44.7%

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

      9. *-commutative 44.7%

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

      10. neg-sub0 44.7%

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

      11. cos-neg 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in im around 0 7.3%

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

      1. neg-mul-1 7.3%

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

      2. unsub-neg 7.3%

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

   7. Simplified 7.3%

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

   8. Taylor expanded in im around 0 59.7%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 59.7%

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

      2. distribute-rgt-out 59.7%

         \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   10. Simplified 59.7%

       \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   if 410 < im < 3.99999999999999992e150

   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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 5.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 5.1%

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

      2. distribute-rgt-out 5.1%

         \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   10. Simplified 5.1%

       \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

   11. Taylor expanded in re around 0 31.7%

       \[\leadsto im \cdot \left(\color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left(-1 + -0.25 \cdot im\right)\right) \]

   if 3.99999999999999992e150 < 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. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 97.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+150}:\\ \;\;\;\;im \cdot \left(\left(-1 + im \cdot -0.25\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   0.5
   (*
    (cos re)
    (*
     im_m
     (-
      (*
       (* im_m im_m)
       (- (* (* im_m im_m) -0.016666666666666666) 0.3333333333333333))
      2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
end

Wolfram

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

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in im around 0 89.7%

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

   1. unpow2 89.7%

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

7. Applied egg-rr 89.7%

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

   1. unpow2 89.7%

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

9. Applied egg-rr 89.7%

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

10. Final simplification 89.7%

    \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right) \]
11. Add Preprocessing

Alternative 8: 87.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   0.5
   (*
    (cos re)
    (*
     im_m
     (-
      (*
       im_m
       (- (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666)) 0.5))
      2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (cos(re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (0.5d0 * (cos(re) * (im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (Math.cos(re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (math.cos(re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (cos(re) * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in im around 0 25.8%

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

   1. neg-mul-1 25.8%

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

   2. unsub-neg 25.8%

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

7. Simplified 25.8%

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

8. Taylor expanded in im around 0 64.0%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \cdot \cos re\right) \]

9. Final simplification 64.0%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 75.4% accurate, 2.8× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* 0.5 (* (cos re) (* im_m (- (* im_m -0.5) 2.0))))))

C

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

Fortran

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

Java

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

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (math.cos(re) * (im_m * ((im_m * -0.5) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * -0.5) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0))));
end

Wolfram

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

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in im around 0 25.8%

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

   1. neg-mul-1 25.8%

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

   2. unsub-neg 25.8%

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

7. Simplified 25.8%

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

8. Taylor expanded in im around 0 59.6%

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

9. Final simplification 59.6%

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

Alternative 10: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(\cos re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* (cos re) (+ -1.0 (* im_m -0.25))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (cos(re) * (-1.0 + (im_m * -0.25))));
}

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 * (cos(re) * ((-1.0d0) + (im_m * (-0.25d0)))))
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 * (Math.cos(re) * (-1.0 + (im_m * -0.25))));
}

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 * (math.cos(re) * (-1.0 + (im_m * -0.25))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(im_m * -0.25)))))
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 * (cos(re) * (-1.0 + (im_m * -0.25))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in im around 0 25.8%

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

   1. neg-mul-1 25.8%

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

   2. unsub-neg 25.8%

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

7. Simplified 25.8%

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

8. Taylor expanded in im around 0 59.6%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + -0.25 \cdot \left(im \cdot \cos re\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 59.6%

      \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(-0.25 \cdot im\right) \cdot \cos re}\right) \]

   2. distribute-rgt-out 59.6%

      \[\leadsto im \cdot \color{blue}{\left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

10. Simplified 59.6%

    \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + -0.25 \cdot im\right)\right)} \]

11. Final simplification 59.6%

    \[\leadsto im \cdot \left(\cos re \cdot \left(-1 + im \cdot -0.25\right)\right) \]
12. Add Preprocessing

Alternative 11: 3.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 0 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * 0.0d0
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

6. Applied egg-rr 3.7%

   \[\leadsto 0.5 \cdot \color{blue}{0} \]

7. Final simplification 3.7%

   \[\leadsto 0 \]
8. Add Preprocessing

Alternative 12: 3.6% accurate, 309.0× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

   1. /-rgt-identity 55.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 55.7%

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

   3. associate-*l/ 55.7%

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

   4. cos-neg 55.7%

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

   5. associate-*l* 55.7%

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

   6. associate-*r/ 55.7%

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

   7. exp-0 55.7%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 55.7%

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

   9. *-commutative 55.7%

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

   10. neg-sub0 55.7%

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

   11. cos-neg 55.7%

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

3. Simplified 55.7%

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

6. Applied egg-rr 2.9%

   \[\leadsto 0.5 \cdot \color{blue}{-2} \]
7. Step-by-step derivation

   1. metadata-eval 2.9%

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

8. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024179 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

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