math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.6%

Time: 8.8s

Alternatives: 10

Speedup: 2.8×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 54.0% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im\_m}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\cos re, im\_m \cdot -2, \left(\cos re \cdot -0.3333333333333333\right) \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -5e+18)
    (* 0.5 (* (- 27.0 (exp im_m)) (cos re)))
    (*
     0.5
     (fma
      (cos re)
      (* im_m -2.0)
      (* (* (cos re) -0.3333333333333333) (pow im_m 3.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -5e+18) {
		tmp = 0.5 * ((27.0 - exp(im_m)) * cos(re));
	} else {
		tmp = 0.5 * fma(cos(re), (im_m * -2.0), ((cos(re) * -0.3333333333333333) * pow(im_m, 3.0)));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -5e+18)
		tmp = Float64(0.5 * Float64(Float64(27.0 - exp(im_m)) * cos(re)));
	else
		tmp = Float64(0.5 * fma(cos(re), Float64(im_m * -2.0), Float64(Float64(cos(re) * -0.3333333333333333) * (im_m ^ 3.0))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -5e+18], N[(0.5 * N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision] + N[(N[(N[Cos[re], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[Power[im$95$m, 3.0], $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)) < -5e18

   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%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   if -5e18 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 94.3%

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

      1. distribute-rgt-in 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*l* 94.3%

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

      4. fma-define 94.3%

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

      5. *-commutative 94.3%

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

      6. associate-*l* 94.3%

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

      7. associate-*r* 94.3%

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

      8. distribute-rgt-out 94.3%

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

      9. +-commutative 94.3%

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

      10. *-commutative 94.3%

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

      11. fma-define 94.3%

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

      12. pow-plus 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 94.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 89.8%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(-0.3333333333333333 \cdot \cos re\right)} \cdot {im}^{3}\right) \]
   9. Step-by-step derivation

      1. *-commutative 89.8%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   10. Simplified 89.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\cos re, im \cdot -2, \left(\cos re \cdot -0.3333333333333333\right) \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im\_m}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \end{array} \end{array} \]

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 (<= (- (exp (- im_m)) (exp im_m)) -5e+18)
    (* 0.5 (* (- 27.0 (exp im_m)) (cos re)))
    (* im_m (* (cos re) (+ (* -0.16666666666666666 (* im_m im_m)) -1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -5e+18) {
		tmp = 0.5 * ((27.0 - exp(im_m)) * cos(re));
	} else {
		tmp = im_m * (cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) - exp(im_m)) <= (-5d+18)) then
        tmp = 0.5d0 * ((27.0d0 - exp(im_m)) * cos(re))
    else
        tmp = im_m * (cos(re) * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -5e+18) {
		tmp = 0.5 * ((27.0 - Math.exp(im_m)) * Math.cos(re));
	} else {
		tmp = im_m * (Math.cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -5e+18:
		tmp = 0.5 * ((27.0 - math.exp(im_m)) * math.cos(re))
	else:
		tmp = im_m * (math.cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -5e+18)
		tmp = Float64(0.5 * Float64(Float64(27.0 - exp(im_m)) * cos(re)));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -5e+18)
		tmp = 0.5 * ((27.0 - exp(im_m)) * cos(re));
	else
		tmp = im_m * (cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -5e+18], N[(0.5 * N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $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)) < -5e18

   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%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   if -5e18 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 94.3%

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

      1. distribute-rgt-in 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*l* 94.3%

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

      4. fma-define 94.3%

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

      5. *-commutative 94.3%

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

      6. associate-*l* 94.3%

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

      7. associate-*r* 94.3%

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

      8. distribute-rgt-out 94.3%

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

      9. +-commutative 94.3%

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

      10. *-commutative 94.3%

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

      11. fma-define 94.3%

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

      12. pow-plus 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 94.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 89.8%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(-0.3333333333333333 \cdot \cos re\right)} \cdot {im}^{3}\right) \]
   9. Step-by-step derivation

      1. *-commutative 89.8%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   10. Simplified 89.8%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   11. Taylor expanded in im around 0 89.8%

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

       1. +-commutative 89.8%

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

       2. associate-*r* 89.8%

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

       3. distribute-rgt-out 89.8%

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

   13. Simplified 89.8%

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

       1. unpow2 89.8%

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

   15. Applied egg-rr 89.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.4 \lor \neg \left(im\_m \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 4.4) (not (<= im_m 1.02e+103)))
    (* im_m (* (cos re) (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))
    (* 0.5 (- 27.0 (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 <= 4.4) || !(im_m <= 1.02e+103)) {
		tmp = im_m * (cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 0.5 * (27.0 - 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 <= 4.4d0) .or. (.not. (im_m <= 1.02d+103))) then
        tmp = im_m * (cos(re) * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    else
        tmp = 0.5d0 * (27.0d0 - 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 <= 4.4) || !(im_m <= 1.02e+103)) {
		tmp = im_m * (Math.cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 0.5 * (27.0 - 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 <= 4.4) or not (im_m <= 1.02e+103):
		tmp = im_m * (math.cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	else:
		tmp = 0.5 * (27.0 - 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 <= 4.4) || !(im_m <= 1.02e+103))
		tmp = Float64(im_m * Float64(cos(re) * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	else
		tmp = Float64(0.5 * Float64(27.0 - 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 <= 4.4) || ~((im_m <= 1.02e+103)))
		tmp = im_m * (cos(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	else
		tmp = 0.5 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 4.4], N[Not[LessEqual[im$95$m, 1.02e+103]], $MachinePrecision]], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.4000000000000004 or 1.01999999999999991e103 < im

   1. Initial program 52.9%

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

      1. /-rgt-identity 52.9%

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

      2. exp-0 52.9%

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

      3. associate-*l/ 52.9%

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

      4. cos-neg 52.9%

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

      5. associate-*l* 52.9%

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

      6. associate-*r/ 52.9%

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

      7. exp-0 52.9%

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

      8. /-rgt-identity 52.9%

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

      9. *-commutative 52.9%

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

      10. neg-sub0 52.9%

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

      11. cos-neg 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in im around 0 95.4%

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

      1. distribute-rgt-in 95.4%

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

      2. *-commutative 95.4%

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

      3. associate-*l* 95.4%

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

      4. fma-define 95.4%

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

      5. *-commutative 95.4%

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

      6. associate-*l* 95.4%

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

      7. associate-*r* 95.4%

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

      8. distribute-rgt-out 95.4%

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

      9. +-commutative 95.4%

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

      10. *-commutative 95.4%

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

      11. fma-define 95.4%

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

      12. pow-plus 95.4%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 95.4%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 95.4%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 91.8%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(-0.3333333333333333 \cdot \cos re\right)} \cdot {im}^{3}\right) \]
   9. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   10. Simplified 91.8%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   11. Taylor expanded in im around 0 91.8%

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

       1. +-commutative 91.8%

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

       2. associate-*r* 91.8%

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

       3. distribute-rgt-out 91.8%

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

   13. Simplified 91.8%

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

       1. unpow2 91.8%

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

   15. Applied egg-rr 91.8%

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

   if 4.4000000000000004 < im < 1.01999999999999991e103

   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%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 82.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(27 - e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4 \lor \neg \left(im \leq 1.02 \cdot 10^{+103}\right):\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.9:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im\_m \cdot \left(im\_m \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

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 3.9)
    (* im_m (- (cos re)))
    (if (<= im_m 1.85e+154)
      (* 0.5 (- 27.0 (exp im_m)))
      (* 0.5 (* (cos re) (+ 26.0 (* im_m (* im_m -0.5)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.9) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.85e+154) {
		tmp = 0.5 * (27.0 - exp(im_m));
	} else {
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.9d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.85d+154) then
        tmp = 0.5d0 * (27.0d0 - exp(im_m))
    else
        tmp = 0.5d0 * (cos(re) * (26.0d0 + (im_m * (im_m * (-0.5d0)))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.9) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.85e+154) {
		tmp = 0.5 * (27.0 - Math.exp(im_m));
	} else {
		tmp = 0.5 * (Math.cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.9:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.85e+154:
		tmp = 0.5 * (27.0 - math.exp(im_m))
	else:
		tmp = 0.5 * (math.cos(re) * (26.0 + (im_m * (im_m * -0.5))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.9)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.85e+154)
		tmp = Float64(0.5 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(26.0 + Float64(im_m * Float64(im_m * -0.5)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.9)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.85e+154)
		tmp = 0.5 * (27.0 - exp(im_m));
	else
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.9], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.85e+154], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(im$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.89999999999999991

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 94.3%

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

      1. distribute-rgt-in 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*l* 94.3%

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

      4. fma-define 94.3%

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

      5. *-commutative 94.3%

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

      6. associate-*l* 94.3%

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

      7. associate-*r* 94.3%

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

      8. distribute-rgt-out 94.3%

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

      9. +-commutative 94.3%

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

      10. *-commutative 94.3%

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

      11. fma-define 94.3%

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

      12. pow-plus 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 94.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. *-commutative 66.2%

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

      3. mul-1-neg 66.2%

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

   10. Simplified 66.2%

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

   if 3.89999999999999991 < im < 1.84999999999999997e154

   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%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 85.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(27 - e^{im}\right)} \]

   if 1.84999999999999997e154 < 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 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right)} \cdot \cos re\right) \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(26 + im \cdot \color{blue}{\left(-0.5 \cdot im\right)}\right) \cdot \cos re\right) \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \left(\left(26 + im \cdot \color{blue}{\left(im \cdot -0.5\right)}\right) \cdot \cos re\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(\left(26 + im \cdot \color{blue}{\left(im \cdot -0.5\right)}\right) \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.9:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im \cdot \left(im \cdot -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 4.0) (* im_m (- (cos re))) (* 0.5 (- 27.0 (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 <= 4.0) {
		tmp = im_m * -cos(re);
	} else {
		tmp = 0.5 * (27.0 - 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 <= 4.0d0) then
        tmp = im_m * -cos(re)
    else
        tmp = 0.5d0 * (27.0d0 - 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 <= 4.0) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = 0.5 * (27.0 - 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 <= 4.0:
		tmp = im_m * -math.cos(re)
	else:
		tmp = 0.5 * (27.0 - 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 <= 4.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64(0.5 * Float64(27.0 - 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 <= 4.0)
		tmp = im_m * -cos(re);
	else
		tmp = 0.5 * (27.0 - 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, 4.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 94.3%

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

      1. distribute-rgt-in 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*l* 94.3%

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

      4. fma-define 94.3%

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

      5. *-commutative 94.3%

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

      6. associate-*l* 94.3%

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

      7. associate-*r* 94.3%

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

      8. distribute-rgt-out 94.3%

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

      9. +-commutative 94.3%

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

      10. *-commutative 94.3%

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

      11. fma-define 94.3%

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

      12. pow-plus 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 94.3%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 94.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. *-commutative 66.2%

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

      3. mul-1-neg 66.2%

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

   10. Simplified 66.2%

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

   if 4 < 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 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 76.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(27 - e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.75e+37)
    (* im_m (- (cos re)))
    (* (pow im_m 3.0) -0.16666666666666666))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.75e+37) {
		tmp = im_m * -cos(re);
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.75d+37) then
        tmp = im_m * -cos(re)
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.75e+37) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.75e+37:
		tmp = im_m * -math.cos(re)
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.75e+37)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.75e+37)
		tmp = im_m * -cos(re);
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.75e+37], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.75e37

   1. Initial program 44.0%

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

      1. /-rgt-identity 44.0%

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

      2. exp-0 44.0%

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

      3. associate-*l/ 44.0%

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

      4. cos-neg 44.0%

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

      5. associate-*l* 44.0%

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

      6. associate-*r/ 44.0%

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

      7. exp-0 44.0%

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

      8. /-rgt-identity 44.0%

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

      9. *-commutative 44.0%

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

      10. neg-sub0 44.0%

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

      11. cos-neg 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in im around 0 90.2%

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

      1. distribute-rgt-in 90.2%

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

      2. *-commutative 90.2%

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

      3. associate-*l* 90.2%

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

      4. fma-define 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.2%

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

      7. associate-*r* 90.2%

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

      8. distribute-rgt-out 90.2%

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

      9. +-commutative 90.2%

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

      10. *-commutative 90.2%

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

      11. fma-define 90.2%

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

      12. pow-plus 90.2%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 90.2%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 90.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 63.3%

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

      1. associate-*r* 63.3%

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

      2. *-commutative 63.3%

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

      3. mul-1-neg 63.3%

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

   10. Simplified 63.3%

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

   if 1.75e37 < 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 92.1%

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

      1. distribute-rgt-in 92.1%

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

      2. *-commutative 92.1%

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

      3. associate-*l* 92.1%

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

      4. fma-define 92.1%

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

      5. *-commutative 92.1%

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

      6. associate-*l* 92.1%

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

      7. associate-*r* 92.1%

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

      8. distribute-rgt-out 92.1%

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

      9. +-commutative 92.1%

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

      10. *-commutative 92.1%

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

      11. fma-define 92.1%

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

      12. pow-plus 92.1%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 92.1%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 92.1%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 79.3%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(-0.3333333333333333 \cdot \cos re\right)} \cdot {im}^{3}\right) \]
   9. Step-by-step derivation

      1. *-commutative 79.3%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   10. Simplified 79.3%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   11. Taylor expanded in re around 0 58.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \]

   12. Taylor expanded in im around inf 58.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.4:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 2.4) (- im_m) (* (pow im_m 3.0) -0.16666666666666666))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.4) {
		tmp = -im_m;
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.4d0) then
        tmp = -im_m
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.4) {
		tmp = -im_m;
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.4:
		tmp = -im_m
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.4)
		tmp = Float64(-im_m);
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.4)
		tmp = -im_m;
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.4], (-im$95$m), N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.39999999999999991

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 66.2%

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

   6. Taylor expanded in re around 0 36.8%

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

      1. mul-1-neg 36.8%

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

   8. Simplified 36.8%

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

   if 2.39999999999999991 < 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 80.5%

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

      1. distribute-rgt-in 80.5%

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

      2. *-commutative 80.5%

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

      3. associate-*l* 80.5%

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

      4. fma-define 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-*l* 80.5%

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

      7. associate-*r* 80.5%

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

      8. distribute-rgt-out 80.5%

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

      9. +-commutative 80.5%

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

      10. *-commutative 80.5%

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

      11. fma-define 80.5%

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

      12. pow-plus 80.5%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot \color{blue}{{im}^{\left(2 + 1\right)}}\right) \]

      13. metadata-eval 80.5%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{\color{blue}{3}}\right) \]

   7. Simplified 80.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\cos re, -2 \cdot im, \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right)\right) \cdot {im}^{3}\right)} \]

   8. Taylor expanded in im around 0 69.4%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(-0.3333333333333333 \cdot \cos re\right)} \cdot {im}^{3}\right) \]
   9. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   10. Simplified 69.4%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\cos re, -2 \cdot im, \color{blue}{\left(\cos re \cdot -0.3333333333333333\right)} \cdot {im}^{3}\right) \]

   11. Taylor expanded in re around 0 51.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)} \]

   12. Taylor expanded in im around inf 51.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.5% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8.2:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;13 + im\_m \cdot \left(im\_m \cdot -0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 8.2) (- im_m) (+ 13.0 (* im_m (- (* im_m -0.25) 0.5))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8.2) {
		tmp = -im_m;
	} else {
		tmp = 13.0 + (im_m * ((im_m * -0.25) - 0.5));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 8.2d0) then
        tmp = -im_m
    else
        tmp = 13.0d0 + (im_m * ((im_m * (-0.25d0)) - 0.5d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8.2) {
		tmp = -im_m;
	} else {
		tmp = 13.0 + (im_m * ((im_m * -0.25) - 0.5));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 8.2:
		tmp = -im_m
	else:
		tmp = 13.0 + (im_m * ((im_m * -0.25) - 0.5))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 8.2)
		tmp = Float64(-im_m);
	else
		tmp = Float64(13.0 + Float64(im_m * Float64(Float64(im_m * -0.25) - 0.5)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 8.2)
		tmp = -im_m;
	else
		tmp = 13.0 + (im_m * ((im_m * -0.25) - 0.5));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 8.2], (-im$95$m), N[(13.0 + N[(im$95$m * N[(N[(im$95$m * -0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 8.1999999999999993

   1. Initial program 41.3%

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

      1. /-rgt-identity 41.3%

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

      2. exp-0 41.3%

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

      3. associate-*l/ 41.3%

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

      4. cos-neg 41.3%

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

      5. associate-*l* 41.3%

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

      6. associate-*r/ 41.3%

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

      7. exp-0 41.3%

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

      8. /-rgt-identity 41.3%

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

      9. *-commutative 41.3%

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

      10. neg-sub0 41.3%

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

      11. cos-neg 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in im around 0 66.2%

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

   6. Taylor expanded in re around 0 36.8%

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

      1. mul-1-neg 36.8%

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

   8. Simplified 36.8%

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

   if 8.1999999999999993 < 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 100.0%

      \[\leadsto 0.5 \cdot \left(\left(\color{blue}{27} - e^{im}\right) \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 63.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right)} \cdot \cos re\right) \]

   8. Taylor expanded in re around 0 45.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right)} \]

   9. Taylor expanded in im around 0 45.8%

      \[\leadsto \color{blue}{13 + im \cdot \left(-0.25 \cdot im - 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8.2:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;13 + im \cdot \left(im \cdot -0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.1% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. /-rgt-identity 56.9%

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

   2. exp-0 56.9%

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

   3. associate-*l/ 56.9%

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

   4. cos-neg 56.9%

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

   5. associate-*l* 56.9%

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

   6. associate-*r/ 56.9%

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

   7. exp-0 56.9%

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

   8. /-rgt-identity 56.9%

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

   9. *-commutative 56.9%

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

   10. neg-sub0 56.9%

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

   11. cos-neg 56.9%

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

3. Simplified 56.9%

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

5. Taylor expanded in im around 0 50.2%

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

6. Taylor expanded in re around 0 28.2%

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

   1. mul-1-neg 28.2%

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

8. Simplified 28.2%

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

Alternative 10: 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 56.9%

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

   1. /-rgt-identity 56.9%

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

   2. exp-0 56.9%

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

   3. associate-*l/ 56.9%

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

   4. cos-neg 56.9%

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

   5. associate-*l* 56.9%

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

   6. associate-*r/ 56.9%

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

   7. exp-0 56.9%

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

   8. /-rgt-identity 56.9%

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

   9. *-commutative 56.9%

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

   10. neg-sub0 56.9%

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

   11. cos-neg 56.9%

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

3. Simplified 56.9%

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

6. Applied egg-rr 2.8%

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

   1. metadata-eval 2.8%

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

8. Applied egg-rr 2.8%

   \[\leadsto \color{blue}{-1} \]
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 2024177 
(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))))