math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.8%

Time: 8.6s

Alternatives: 11

Speedup: 2.8×

• 49.4% 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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.05)
      (* 0.5 (* (cos re) t_0))
      (* im_m (* (cos re) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = 0.5 * (cos(re) * t_0);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.05d0)) then
        tmp = 0.5d0 * (cos(re) * t_0)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = 0.5 * (Math.cos(re) * t_0);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.05:
		tmp = 0.5 * (math.cos(re) * t_0)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(0.5 * Float64(cos(re) * t_0));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = 0.5 * (cos(re) * t_0);
	else
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.05], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $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)) < -0.050000000000000003

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 94.2%

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

      1. distribute-rgt-in 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*l* 94.2%

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

      4. fma-define 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*l* 94.2%

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

      7. associate-*r* 94.2%

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

      8. distribute-rgt-out 94.2%

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

      9. +-commutative 94.2%

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

      10. *-commutative 94.2%

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

      11. fma-define 94.2%

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

      12. pow-plus 94.2%

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

      13. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in im around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. distribute-rgt-out 87.1%

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

      3. *-commutative 87.1%

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

   10. Simplified 87.1%

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

       1. unpow2 87.1%

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

   12. Applied egg-rr 87.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im\_m}^{2}, -0.16666666666666666, -1\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   im_m
   (log1p
    (expm1 (* (cos re) (fma (pow im_m 2.0) -0.16666666666666666 -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 * (im_m * log1p(expm1((cos(re) * fma(pow(im_m, 2.0), -0.16666666666666666, -1.0)))));
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * log1p(expm1(Float64(cos(re) * fma((im_m ^ 2.0), -0.16666666666666666, -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 * N[(im$95$m * N[Log[1 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

   1. /-rgt-identity 54.1%

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

   2. exp-0 54.1%

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

   3. associate-*l/ 54.1%

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

   4. cos-neg 54.1%

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

   5. associate-*l* 54.1%

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

   6. associate-*r/ 54.1%

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

   7. exp-0 54.1%

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

   8. /-rgt-identity 54.1%

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

   9. *-commutative 54.1%

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

   10. neg-sub0 54.1%

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

   11. cos-neg 54.1%

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

3. Simplified 54.1%

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

5. Taylor expanded in im around 0 91.3%

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

   1. distribute-rgt-in 91.3%

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

   2. *-commutative 91.3%

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

   3. associate-*l* 91.3%

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

   4. fma-define 91.3%

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

   5. *-commutative 91.3%

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

   6. associate-*l* 91.3%

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

   7. associate-*r* 91.3%

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

   8. distribute-rgt-out 91.3%

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

   9. +-commutative 91.3%

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

   10. *-commutative 91.3%

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

   11. fma-define 91.3%

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

   12. pow-plus 91.3%

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

   13. metadata-eval 91.3%

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

7. Simplified 91.3%

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

8. Taylor expanded in im around 0 84.4%

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

   1. associate-*r* 84.4%

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

   2. distribute-rgt-out 84.4%

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

   3. *-commutative 84.4%

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

10. Simplified 84.4%

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

    1. log1p-expm1-u 99.5%

       \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

    2. +-commutative 99.5%

       \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

    3. fma-define 99.5%

       \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

12. Applied egg-rr 99.5%

    \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]
13. Add Preprocessing

Alternative 3: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.055 \lor \neg \left(im\_m \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 0.055) (not (<= im_m 1.05e+103)))
    (* im_m (* (cos re) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (* 0.5 (- (exp (- im_m)) (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.055) || !(im_m <= 1.05e+103)) {
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 0.055d0) .or. (.not. (im_m <= 1.05d+103))) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.055) || !(im_m <= 1.05e+103)) {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 0.055) or not (im_m <= 1.05e+103):
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	else:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 0.055) || !(im_m <= 1.05e+103))
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 0.055) || ~((im_m <= 1.05e+103)))
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	else
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 0.055], N[Not[LessEqual[im$95$m, 1.05e+103]], $MachinePrecision]], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.0550000000000000003 or 1.0500000000000001e103 < im

   1. Initial program 51.1%

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

      1. /-rgt-identity 51.1%

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

      2. exp-0 51.1%

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

      3. associate-*l/ 51.1%

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

      4. cos-neg 51.1%

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

      5. associate-*l* 51.1%

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

      6. associate-*r/ 51.1%

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

      7. exp-0 51.1%

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

      8. /-rgt-identity 51.1%

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

      9. *-commutative 51.1%

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

      10. neg-sub0 51.1%

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

      11. cos-neg 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in im around 0 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 89.7%

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

      1. associate-*r* 89.7%

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

      2. distribute-rgt-out 89.7%

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

      3. *-commutative 89.7%

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

   10. Simplified 89.7%

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

       1. unpow2 89.7%

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

   12. Applied egg-rr 89.7%

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

   if 0.0550000000000000003 < im < 1.0500000000000001e103

   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 re around 0 81.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.055 \lor \neg \left(im \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.052:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.052)
    (* im_m (* (cos re) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (if (<= im_m 4.5e+61)
      (* 0.5 (- (exp (- im_m)) (exp im_m)))
      (* (cos re) (* -0.008333333333333333 (pow im_m 5.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.052) {
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	} else {
		tmp = cos(re) * (-0.008333333333333333 * pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.052d0) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else if (im_m <= 4.5d+61) then
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    else
        tmp = cos(re) * ((-0.008333333333333333d0) * (im_m ** 5.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.052) {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (-0.008333333333333333 * Math.pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.052:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	elif im_m <= 4.5e+61:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	else:
		tmp = math.cos(re) * (-0.008333333333333333 * math.pow(im_m, 5.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.052)
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64(-0.008333333333333333 * (im_m ^ 5.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.052)
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	elseif (im_m <= 4.5e+61)
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	else
		tmp = cos(re) * (-0.008333333333333333 * (im_m ^ 5.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.052], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0519999999999999976

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 94.2%

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

      1. distribute-rgt-in 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*l* 94.2%

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

      4. fma-define 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*l* 94.2%

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

      7. associate-*r* 94.2%

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

      8. distribute-rgt-out 94.2%

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

      9. +-commutative 94.2%

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

      10. *-commutative 94.2%

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

      11. fma-define 94.2%

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

      12. pow-plus 94.2%

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

      13. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in im around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. distribute-rgt-out 87.1%

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

      3. *-commutative 87.1%

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

   10. Simplified 87.1%

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

       1. unpow2 87.1%

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

   12. Applied egg-rr 87.1%

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

   if 0.0519999999999999976 < im < 4.5e61

   1. Initial program 100.0%

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

      1. /-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 re around 0 83.3%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-define 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-out 100.0%

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

      9. +-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. fma-define 100.0%

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

      12. pow-plus 100.0%

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

      13. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.052:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(27 - e^{im\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.0)
    (* im_m (* (cos re) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (* 0.5 (* (cos re) (- 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) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = 0.5 * (cos(re) * (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) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else
        tmp = 0.5d0 * (cos(re) * (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) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = 0.5 * (Math.cos(re) * (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) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	else:
		tmp = 0.5 * (math.cos(re) * (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) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * 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) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	else
		tmp = 0.5 * (cos(re) * (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[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 94.2%

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

      1. distribute-rgt-in 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*l* 94.2%

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

      4. fma-define 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*l* 94.2%

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

      7. associate-*r* 94.2%

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

      8. distribute-rgt-out 94.2%

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

      9. +-commutative 94.2%

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

      10. *-commutative 94.2%

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

      11. fma-define 94.2%

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

      12. pow-plus 94.2%

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

      13. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in im around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. distribute-rgt-out 87.1%

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

      3. *-commutative 87.1%

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

   10. Simplified 87.1%

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

       1. unpow2 87.1%

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

   12. Applied egg-rr 87.1%

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

   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%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(27 - e^{im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 550000 \lor \neg \left(im\_m \leq 1.2 \cdot 10^{+103}\right):\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 550000.0) (not (<= im_m 1.2e+103)))
    (* im_m (* (cos re) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (* 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 <= 550000.0) || !(im_m <= 1.2e+103)) {
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} 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 <= 550000.0d0) .or. (.not. (im_m <= 1.2d+103))) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    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 <= 550000.0) || !(im_m <= 1.2e+103)) {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} 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 <= 550000.0) or not (im_m <= 1.2e+103):
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	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 <= 550000.0) || !(im_m <= 1.2e+103))
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	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 <= 550000.0) || ~((im_m <= 1.2e+103)))
		tmp = im_m * (cos(re) * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	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, 550000.0], N[Not[LessEqual[im$95$m, 1.2e+103]], $MachinePrecision]], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $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 < 5.5e5 or 1.1999999999999999e103 < im

   1. Initial program 51.1%

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

      1. /-rgt-identity 51.1%

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

      2. exp-0 51.1%

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

      3. associate-*l/ 51.1%

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

      4. cos-neg 51.1%

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

      5. associate-*l* 51.1%

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

      6. associate-*r/ 51.1%

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

      7. exp-0 51.1%

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

      8. /-rgt-identity 51.1%

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

      9. *-commutative 51.1%

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

      10. neg-sub0 51.1%

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

      11. cos-neg 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in im around 0 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 89.7%

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

      1. associate-*r* 89.7%

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

      2. distribute-rgt-out 89.7%

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

      3. *-commutative 89.7%

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

   10. Simplified 89.7%

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

       1. unpow2 89.7%

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

   12. Applied egg-rr 89.7%

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

   if 5.5e5 < im < 1.1999999999999999e103

   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 81.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550000 \lor \neg \left(im \leq 1.2 \cdot 10^{+103}\right):\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 550000:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 550000.0) (* (cos re) (- im_m)) (* 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 <= 550000.0) {
		tmp = cos(re) * -im_m;
	} 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 <= 550000.0d0) then
        tmp = cos(re) * -im_m
    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 <= 550000.0) {
		tmp = Math.cos(re) * -im_m;
	} 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 <= 550000.0:
		tmp = math.cos(re) * -im_m
	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 <= 550000.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	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 <= 550000.0)
		tmp = cos(re) * -im_m;
	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, 550000.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $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 < 5.5e5

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 94.2%

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

      1. distribute-rgt-in 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-*l* 94.2%

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

      4. fma-define 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*l* 94.2%

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

      7. associate-*r* 94.2%

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

      8. distribute-rgt-out 94.2%

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

      9. +-commutative 94.2%

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

      10. *-commutative 94.2%

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

      11. fma-define 94.2%

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

      12. pow-plus 94.2%

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

      13. metadata-eval 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in im around 0 68.1%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

      3. mul-1-neg 68.1%

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

   10. Simplified 68.1%

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

   if 5.5e5 < 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 82.8%

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

Alternative 8: 69.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.8 \cdot 10^{+84}:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im\_m \cdot \left(-1 + im\_m \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.8e+84)
    (* (cos re) (- im_m))
    (* 0.5 (+ 26.0 (* im_m (+ -1.0 (* im_m -0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.8e+84) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 2.8d+84) then
        tmp = cos(re) * -im_m
    else
        tmp = 0.5d0 * (26.0d0 + (im_m * ((-1.0d0) + (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 <= 2.8e+84) {
		tmp = Math.cos(re) * -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 2.8e+84:
		tmp = math.cos(re) * -im_m
	else:
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 2.8e+84)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(0.5 * Float64(26.0 + Float64(im_m * Float64(-1.0 + 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 <= 2.8e+84)
		tmp = cos(re) * -im_m;
	else
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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, 2.8e+84], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(0.5 * N[(26.0 + N[(im$95$m * N[(-1.0 + N[(im$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.79999999999999982e84

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 89.1%

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

      1. distribute-rgt-in 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 89.1%

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

      4. fma-define 89.1%

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

      5. *-commutative 89.1%

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

      6. associate-*l* 89.1%

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

      7. associate-*r* 89.1%

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

      8. distribute-rgt-out 89.1%

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

      9. +-commutative 89.1%

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

      10. *-commutative 89.1%

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

      11. fma-define 89.1%

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

      12. pow-plus 89.1%

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

      13. metadata-eval 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in im around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. *-commutative 64.0%

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

      3. mul-1-neg 64.0%

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

   10. Simplified 64.0%

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

   if 2.79999999999999982e84 < 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 58.3%

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

   8. Taylor expanded in re around 0 48.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.8 \cdot 10^{+84}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im \cdot \left(-1 + im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.1% accurate, 19.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 550000:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im\_m \cdot \left(-1 + im\_m \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

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 550000.0)
    (- im_m)
    (* 0.5 (+ 26.0 (* im_m (+ -1.0 (* im_m -0.5))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 550000.0) {
		tmp = -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 550000.0d0) then
        tmp = -im_m
    else
        tmp = 0.5d0 * (26.0d0 + (im_m * ((-1.0d0) + (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 <= 550000.0) {
		tmp = -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 550000.0:
		tmp = -im_m
	else:
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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 <= 550000.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(0.5 * Float64(26.0 + Float64(im_m * Float64(-1.0 + 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 <= 550000.0)
		tmp = -im_m;
	else
		tmp = 0.5 * (26.0 + (im_m * (-1.0 + (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, 550000.0], (-im$95$m), N[(0.5 * N[(26.0 + N[(im$95$m * N[(-1.0 + N[(im$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.5e5

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 68.1%

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

   6. Taylor expanded in re around 0 40.5%

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

      1. mul-1-neg 40.5%

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

   8. Simplified 40.5%

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

   if 5.5e5 < 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 47.2%

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

   8. Taylor expanded in re around 0 39.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im \cdot \left(-1 + im \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.3% 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 54.1%

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

   1. /-rgt-identity 54.1%

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

   2. exp-0 54.1%

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

   3. associate-*l/ 54.1%

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

   4. cos-neg 54.1%

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

   5. associate-*l* 54.1%

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

   6. associate-*r/ 54.1%

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

   7. exp-0 54.1%

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

   8. /-rgt-identity 54.1%

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

   9. *-commutative 54.1%

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

   10. neg-sub0 54.1%

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

   11. cos-neg 54.1%

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

3. Simplified 54.1%

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

5. Taylor expanded in im around 0 52.5%

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

6. Taylor expanded in re around 0 31.6%

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

   1. mul-1-neg 31.6%

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

8. Simplified 31.6%

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

Alternative 11: 3.5% 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 54.1%

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

   1. /-rgt-identity 54.1%

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

   2. exp-0 54.1%

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

   3. associate-*l/ 54.1%

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

   4. cos-neg 54.1%

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

   5. associate-*l* 54.1%

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

   6. associate-*r/ 54.1%

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

   7. exp-0 54.1%

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

   8. /-rgt-identity 54.1%

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

   9. *-commutative 54.1%

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

   10. neg-sub0 54.1%

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

   11. cos-neg 54.1%

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

3. Simplified 54.1%

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

6. Applied egg-rr 3.0%

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

   1. metadata-eval 3.0%

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

8. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-1} \]
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 2024181 
(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))))