math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.5%

Time: 10.6s

Alternatives: 16

Speedup: 2.8×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\cos re \cdot \mathsf{fma}\left({im\_m}^{2}, -0.016666666666666666, -0.3333333333333333\right), {im\_m}^{3}, \cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -1e+26)
      (* 0.5 (* t_0 (cos re)))
      (*
       0.5
       (fma
        (*
         (cos re)
         (fma (pow im_m 2.0) -0.016666666666666666 -0.3333333333333333))
        (pow im_m 3.0)
        (* (cos re) (* im_m -2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -1e+26) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = 0.5 * fma((cos(re) * fma(pow(im_m, 2.0), -0.016666666666666666, -0.3333333333333333)), pow(im_m, 3.0), (cos(re) * (im_m * -2.0)));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -1e+26)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(0.5 * fma(Float64(cos(re) * fma((im_m ^ 2.0), -0.016666666666666666, -0.3333333333333333)), (im_m ^ 3.0), Float64(cos(re) * Float64(im_m * -2.0))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := 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, -1e+26], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Power[im$95$m, 3.0], $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -1.00000000000000005e26

   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 -1.00000000000000005e26 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. exp-0 42.3%

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

      3. associate-*l/ 42.3%

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

      4. cos-neg 42.3%

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

      5. associate-*l* 42.3%

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

      6. associate-*r/ 42.3%

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

      7. exp-0 42.3%

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

      8. /-rgt-identity 42.3%

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

      9. *-commutative 42.3%

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

      10. neg-sub0 42.3%

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

      11. cos-neg 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in im around 0 90.8%

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

      1. distribute-lft-in 90.8%

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

      2. +-commutative 90.8%

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

      3. associate-*r* 90.8%

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

      4. *-commutative 90.8%

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

      5. fma-undefine 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), {im}^{3}, \cos re \cdot \left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -1e+26)
      (* 0.5 (* t_0 (cos re)))
      (*
       0.5
       (*
        (cos re)
        (*
         im_m
         (-
          (*
           (pow im_m 2.0)
           (- (* (pow im_m 2.0) -0.016666666666666666) 0.3333333333333333))
          2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -1e+26) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-1d+26)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -1e+26) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -1e+26:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -1e+26)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -1e+26)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = 0.5 * (cos(re) * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.016666666666666666) - 0.3333333333333333)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -1e+26], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -1.00000000000000005e26

   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 -1.00000000000000005e26 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. exp-0 42.3%

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

      3. associate-*l/ 42.3%

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

      4. cos-neg 42.3%

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

      5. associate-*l* 42.3%

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

      6. associate-*r/ 42.3%

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

      7. exp-0 42.3%

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

      8. /-rgt-identity 42.3%

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

      9. *-commutative 42.3%

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

      10. neg-sub0 42.3%

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

      11. cos-neg 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in im around 0 90.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left({im\_m}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -1e+26)
      (* 0.5 (* t_0 (cos re)))
      (*
       0.5
       (*
        im_m
        (* (cos re) (- (* (pow im_m 2.0) -0.3333333333333333) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -1e+26) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = 0.5 * (im_m * (cos(re) * ((pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-1d+26)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = 0.5d0 * (im_m * (cos(re) * (((im_m ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -1e+26) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((Math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -1e+26:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = 0.5 * (im_m * (math.cos(re) * ((math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -1e+26)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64((im_m ^ 2.0) * -0.3333333333333333) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -1e+26)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = 0.5 * (im_m * (cos(re) * (((im_m ^ 2.0) * -0.3333333333333333) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -1e+26], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -1.00000000000000005e26

   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 -1.00000000000000005e26 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. exp-0 42.3%

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

      3. associate-*l/ 42.3%

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

      4. cos-neg 42.3%

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

      5. associate-*l* 42.3%

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

      6. associate-*r/ 42.3%

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

      7. exp-0 42.3%

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

      8. /-rgt-identity 42.3%

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

      9. *-commutative 42.3%

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

      10. neg-sub0 42.3%

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

      11. cos-neg 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in im around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. distribute-rgt-out 85.3%

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

      3. +-commutative 85.3%

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

      4. metadata-eval 85.3%

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

      5. sub-neg 85.3%

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

      6. *-commutative 85.3%

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

      7. fmm-def 85.3%

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

      8. metadata-eval 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in im around 0 85.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10000000:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left({im\_m}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.3333333333333333 \cdot {im\_m}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im\_m}^{5}\right)\right)\\ \end{array} \end{array} \]

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 10000000.0)
    (*
     0.5
     (* im_m (* (cos re) (- (* (pow im_m 2.0) -0.3333333333333333) 2.0))))
    (if (<= im_m 4.5e+61)
      (* 0.5 (log1p (expm1 (* -0.3333333333333333 (pow im_m 3.0)))))
      (* 0.5 (* -0.016666666666666666 (* (cos re) (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 <= 10000000.0) {
		tmp = 0.5 * (im_m * (cos(re) * ((pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((-0.3333333333333333 * pow(im_m, 3.0))));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * pow(im_m, 5.0)));
	}
	return im_s * tmp;
}

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 <= 10000000.0) {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((Math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((-0.3333333333333333 * Math.pow(im_m, 3.0))));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * 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 <= 10000000.0:
		tmp = 0.5 * (im_m * (math.cos(re) * ((math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)))
	elif im_m <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((-0.3333333333333333 * math.pow(im_m, 3.0))))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * 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 <= 10000000.0)
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64((im_m ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(-0.3333333333333333 * (im_m ^ 3.0)))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (im_m ^ 5.0))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 10000000.0], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 1e7

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 84.9%

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

      1. associate-*r* 84.9%

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

      2. distribute-rgt-out 84.9%

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

      3. +-commutative 84.9%

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

      4. metadata-eval 84.9%

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

      5. sub-neg 84.9%

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

      6. *-commutative 84.9%

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

      7. fmm-def 84.9%

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

      8. metadata-eval 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in im around 0 84.9%

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

   if 1e7 < 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 im around 0 4.1%

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

      1. associate-*r* 4.1%

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

      2. distribute-rgt-out 4.1%

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

      3. +-commutative 4.1%

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

      4. metadata-eval 4.1%

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

      5. sub-neg 4.1%

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

      6. *-commutative 4.1%

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

      7. fmm-def 4.1%

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

      8. metadata-eval 4.1%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in re around 0 3.0%

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

   9. Taylor expanded in im around inf 3.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 70.0%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   11. Applied egg-rr 70.0%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   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-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 700:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left({im\_m}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(-0.016666666666666666 \cdot {im\_m}^{4} - 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 700.0)
    (*
     0.5
     (* im_m (* (cos re) (- (* (pow im_m 2.0) -0.3333333333333333) 2.0))))
    (if (<= im_m 3.3e+51)
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (*
       0.5
       (*
        (cos re)
        (* im_m (- (* -0.016666666666666666 (pow im_m 4.0)) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 700.0) {
		tmp = 0.5 * (im_m * (cos(re) * ((pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 3.3e+51) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((-0.016666666666666666 * pow(im_m, 4.0)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 700.0d0) then
        tmp = 0.5d0 * (im_m * (cos(re) * (((im_m ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0)))
    else if (im_m <= 3.3d+51) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * (((-0.016666666666666666d0) * (im_m ** 4.0d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 700.0) {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((Math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 3.3e+51) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((-0.016666666666666666 * Math.pow(im_m, 4.0)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 700.0:
		tmp = 0.5 * (im_m * (math.cos(re) * ((math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)))
	elif im_m <= 3.3e+51:
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((-0.016666666666666666 * math.pow(im_m, 4.0)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 700.0)
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64((im_m ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im_m <= 3.3e+51)
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(-0.016666666666666666 * (im_m ^ 4.0)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 700.0)
		tmp = 0.5 * (im_m * (cos(re) * (((im_m ^ 2.0) * -0.3333333333333333) - 2.0)));
	elseif (im_m <= 3.3e+51)
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (cos(re) * (im_m * ((-0.016666666666666666 * (im_m ^ 4.0)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 700.0], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.3e+51], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(-0.016666666666666666 * N[Power[im$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 700

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 84.9%

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

      1. associate-*r* 84.9%

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

      2. distribute-rgt-out 84.9%

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

      3. +-commutative 84.9%

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

      4. metadata-eval 84.9%

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

      5. sub-neg 84.9%

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

      6. *-commutative 84.9%

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

      7. fmm-def 84.9%

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

      8. metadata-eval 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in im around 0 84.9%

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

   if 700 < im < 3.2999999999999997e51

   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 3.3%

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

   6. Taylor expanded in re around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. distribute-lft-out 43.4%

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

   8. Simplified 43.4%

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

   if 3.2999999999999997e51 < 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 95.2%

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

   6. Taylor expanded in im around inf 95.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.016666666666666666 \cdot {im}^{4}} - 2\right)\right) \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.016666666666666666 \cdot {im}^{4} - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 650:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im\_m}^{5}\right)\right)\\ \end{array} \end{array} \]

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 650.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (<= im_m 3.3e+51)
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (* 0.5 (* -0.016666666666666666 (* (cos re) (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 <= 650.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if (im_m <= 3.3e+51) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 650.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if (im_m <= 3.3d+51) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (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 <= 650.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if (im_m <= 3.3e+51) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * 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 <= 650.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif im_m <= 3.3e+51:
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * 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 <= 650.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif (im_m <= 3.3e+51)
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (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 <= 650.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif (im_m <= 3.3e+51)
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (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, 650.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.3e+51], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 650

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 64.7%

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

   if 650 < im < 3.2999999999999997e51

   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 3.3%

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

   6. Taylor expanded in re around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. distribute-lft-out 43.4%

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

   8. Simplified 43.4%

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

   if 3.2999999999999997e51 < 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 95.2%

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

      1. distribute-lft-in 95.2%

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

      2. +-commutative 95.2%

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

      3. associate-*r* 95.2%

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

      4. *-commutative 95.2%

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

      5. fma-undefine 95.2%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in im around inf 95.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 650:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(\cos re \cdot \left({im\_m}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im\_m}^{5}\right)\right)\\ \end{array} \end{array} \]

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 650.0)
    (*
     0.5
     (* im_m (* (cos re) (- (* (pow im_m 2.0) -0.3333333333333333) 2.0))))
    (if (<= im_m 2e+51)
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (* 0.5 (* -0.016666666666666666 (* (cos re) (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 <= 650.0) {
		tmp = 0.5 * (im_m * (cos(re) * ((pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 2e+51) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 650.0d0) then
        tmp = 0.5d0 * (im_m * (cos(re) * (((im_m ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0)))
    else if (im_m <= 2d+51) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (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 <= 650.0) {
		tmp = 0.5 * (im_m * (Math.cos(re) * ((Math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im_m <= 2e+51) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * 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 <= 650.0:
		tmp = 0.5 * (im_m * (math.cos(re) * ((math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0)))
	elif im_m <= 2e+51:
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * 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 <= 650.0)
		tmp = Float64(0.5 * Float64(im_m * Float64(cos(re) * Float64(Float64((im_m ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im_m <= 2e+51)
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (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 <= 650.0)
		tmp = 0.5 * (im_m * (cos(re) * (((im_m ^ 2.0) * -0.3333333333333333) - 2.0)));
	elseif (im_m <= 2e+51)
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (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, 650.0], N[(0.5 * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2e+51], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 650

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 84.9%

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

      1. associate-*r* 84.9%

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

      2. distribute-rgt-out 84.9%

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

      3. +-commutative 84.9%

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

      4. metadata-eval 84.9%

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

      5. sub-neg 84.9%

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

      6. *-commutative 84.9%

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

      7. fmm-def 84.9%

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

      8. metadata-eval 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in im around 0 84.9%

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

   if 650 < im < 2e51

   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 3.3%

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

   6. Taylor expanded in re around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. distribute-lft-out 43.4%

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

   8. Simplified 43.4%

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

   if 2e51 < 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 95.2%

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

      1. distribute-lft-in 95.2%

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

      2. +-commutative 95.2%

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

      3. associate-*r* 95.2%

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

      4. *-commutative 95.2%

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

      5. fma-undefine 95.2%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in im around inf 95.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 520:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+97} \lor \neg \left(im\_m \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 520.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (or (<= im_m 2.8e+97) (not (<= im_m 1.12e+282)))
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (* 0.5 (* im_m (- (* (pow im_m 2.0) -0.3333333333333333) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 520.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (im_m * ((pow(im_m, 2.0) * -0.3333333333333333) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 520.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if ((im_m <= 2.8d+97) .or. (.not. (im_m <= 1.12d+282))) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (im_m * (((im_m ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 520.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (im_m * ((Math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 520.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif (im_m <= 2.8e+97) or not (im_m <= 1.12e+282):
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (im_m * ((math.pow(im_m, 2.0) * -0.3333333333333333) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 520.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282))
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * -0.3333333333333333) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 520.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif ((im_m <= 2.8e+97) || ~((im_m <= 1.12e+282)))
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (im_m * (((im_m ^ 2.0) * -0.3333333333333333) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 520.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 2.8e+97], N[Not[LessEqual[im$95$m, 1.12e+282]], $MachinePrecision]], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 520

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 64.7%

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

   if 520 < im < 2.7999999999999999e97 or 1.12e282 < 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 9.0%

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

   6. Taylor expanded in re around 0 47.5%

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

      1. *-commutative 47.5%

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

      2. distribute-lft-out 47.5%

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

   8. Simplified 47.5%

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

   if 2.7999999999999999e97 < im < 1.12e282

   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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. fmm-def 100.0%

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

      8. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 76.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 520:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+97} \lor \neg \left(im \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 650:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+67} \lor \neg \left(im\_m \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.016666666666666666 \cdot {im\_m}^{4} - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 650.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (or (<= im_m 2e+67) (not (<= im_m 1.12e+282)))
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (* 0.5 (* im_m (- (* -0.016666666666666666 (pow im_m 4.0)) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 650.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2e+67) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (im_m * ((-0.016666666666666666 * pow(im_m, 4.0)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 650.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if ((im_m <= 2d+67) .or. (.not. (im_m <= 1.12d+282))) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * (im_m * (((-0.016666666666666666d0) * (im_m ** 4.0d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 650.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2e+67) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (im_m * ((-0.016666666666666666 * Math.pow(im_m, 4.0)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 650.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif (im_m <= 2e+67) or not (im_m <= 1.12e+282):
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (im_m * ((-0.016666666666666666 * math.pow(im_m, 4.0)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 650.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif ((im_m <= 2e+67) || !(im_m <= 1.12e+282))
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(-0.016666666666666666 * (im_m ^ 4.0)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 650.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif ((im_m <= 2e+67) || ~((im_m <= 1.12e+282)))
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (im_m * ((-0.016666666666666666 * (im_m ^ 4.0)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 650.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 2e+67], N[Not[LessEqual[im$95$m, 1.12e+282]], $MachinePrecision]], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(-0.016666666666666666 * N[Power[im$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 650

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 64.7%

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

   if 650 < im < 1.99999999999999997e67 or 1.12e282 < 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 10.3%

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

   6. Taylor expanded in re around 0 53.3%

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

      1. *-commutative 53.3%

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

      2. distribute-lft-out 53.3%

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

   8. Simplified 53.3%

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

   if 1.99999999999999997e67 < im < 1.12e282

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.016666666666666666 \cdot {im}^{4}} - 2\right)\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 72.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.016666666666666666 \cdot {im}^{4} - 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+67} \lor \neg \left(im \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.016666666666666666 \cdot {im}^{4} - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 550:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+97} \lor \neg \left(im\_m \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 550.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (or (<= im_m 2.8e+97) (not (<= im_m 1.12e+282)))
      (* 0.5 (* im_m (+ -2.0 (pow re 2.0))))
      (* 0.5 (* -0.3333333333333333 (pow im_m 3.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 550.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * pow(im_m, 3.0));
	}
	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 <= 550.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if ((im_m <= 2.8d+97) .or. (.not. (im_m <= 1.12d+282))) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + (re ** 2.0d0)))
    else
        tmp = 0.5d0 * ((-0.3333333333333333d0) * (im_m ** 3.0d0))
    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 <= 550.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282)) {
		tmp = 0.5 * (im_m * (-2.0 + Math.pow(re, 2.0)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * Math.pow(im_m, 3.0));
	}
	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 <= 550.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif (im_m <= 2.8e+97) or not (im_m <= 1.12e+282):
		tmp = 0.5 * (im_m * (-2.0 + math.pow(re, 2.0)))
	else:
		tmp = 0.5 * (-0.3333333333333333 * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 550.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif ((im_m <= 2.8e+97) || !(im_m <= 1.12e+282))
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + (re ^ 2.0))));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * (im_m ^ 3.0)));
	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 <= 550.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif ((im_m <= 2.8e+97) || ~((im_m <= 1.12e+282)))
		tmp = 0.5 * (im_m * (-2.0 + (re ^ 2.0)));
	else
		tmp = 0.5 * (-0.3333333333333333 * (im_m ^ 3.0));
	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, 550.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 2.8e+97], N[Not[LessEqual[im$95$m, 1.12e+282]], $MachinePrecision]], N[(0.5 * N[(im$95$m * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 550

   1. Initial program 42.6%

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

      1. /-rgt-identity 42.6%

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

      2. exp-0 42.6%

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

      3. associate-*l/ 42.6%

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

      4. cos-neg 42.6%

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

      5. associate-*l* 42.6%

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

      6. associate-*r/ 42.6%

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

      7. exp-0 42.6%

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

      8. /-rgt-identity 42.6%

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

      9. *-commutative 42.6%

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

      10. neg-sub0 42.6%

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

      11. cos-neg 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in im around 0 64.7%

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

   if 550 < im < 2.7999999999999999e97 or 1.12e282 < 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 9.0%

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

   6. Taylor expanded in re around 0 47.5%

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

      1. *-commutative 47.5%

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

      2. distribute-lft-out 47.5%

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

   8. Simplified 47.5%

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

   if 2.7999999999999999e97 < im < 1.12e282

   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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. fmm-def 100.0%

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

      8. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 76.9%

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

   9. Taylor expanded in im around inf 76.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+97} \lor \neg \left(im \leq 1.12 \cdot 10^{+282}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.15 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.15e+67)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (* 0.5 (* -0.3333333333333333 (pow im_m 3.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.15e+67) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.15d+67) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else
        tmp = 0.5d0 * ((-0.3333333333333333d0) * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.15e+67) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.15e+67:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	else:
		tmp = 0.5 * (-0.3333333333333333 * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.15e+67)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.15e+67)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	else
		tmp = 0.5 * (-0.3333333333333333 * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.15e+67], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.1500000000000001e67

   1. Initial program 46.0%

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

      1. /-rgt-identity 46.0%

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

      2. exp-0 46.0%

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

      3. associate-*l/ 46.0%

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

      4. cos-neg 46.0%

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

      5. associate-*l* 46.0%

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

      6. associate-*r/ 46.0%

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

      7. exp-0 46.0%

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

      8. /-rgt-identity 46.0%

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

      9. *-commutative 46.0%

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

      10. neg-sub0 46.0%

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

      11. cos-neg 46.0%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in im around 0 61.1%

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

   if 2.1500000000000001e67 < 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 91.2%

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

      1. associate-*r* 91.2%

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

      2. distribute-rgt-out 91.2%

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

      3. +-commutative 91.2%

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

      4. metadata-eval 91.2%

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

      5. sub-neg 91.2%

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

      6. *-commutative 91.2%

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

      7. fmm-def 91.2%

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

      8. metadata-eval 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in re around 0 60.6%

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

   9. Taylor expanded in im around inf 60.6%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.15 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.25:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.25)
    (* 0.5 (* im_m -2.0))
    (* 0.5 (* -0.3333333333333333 (pow im_m 3.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.25) {
		tmp = 0.5 * (im_m * -2.0);
	} else {
		tmp = 0.5 * (-0.3333333333333333 * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.25d0) then
        tmp = 0.5d0 * (im_m * (-2.0d0))
    else
        tmp = 0.5d0 * ((-0.3333333333333333d0) * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.25) {
		tmp = 0.5 * (im_m * -2.0);
	} else {
		tmp = 0.5 * (-0.3333333333333333 * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.25:
		tmp = 0.5 * (im_m * -2.0)
	else:
		tmp = 0.5 * (-0.3333333333333333 * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.25)
		tmp = Float64(0.5 * Float64(im_m * -2.0));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.25)
		tmp = 0.5 * (im_m * -2.0);
	else
		tmp = 0.5 * (-0.3333333333333333 * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.25], N[(0.5 * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.25

   1. Initial program 42.3%

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

      1. /-rgt-identity 42.3%

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

      2. exp-0 42.3%

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

      3. associate-*l/ 42.3%

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

      4. cos-neg 42.3%

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

      5. associate-*l* 42.3%

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

      6. associate-*r/ 42.3%

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

      7. exp-0 42.3%

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

      8. /-rgt-identity 42.3%

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

      9. *-commutative 42.3%

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

      10. neg-sub0 42.3%

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

      11. cos-neg 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in im around 0 65.0%

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

   6. Taylor expanded in re around 0 37.8%

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

   if 2.25 < 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 74.2%

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

      1. associate-*r* 74.2%

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

      2. distribute-rgt-out 74.2%

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

      3. +-commutative 74.2%

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

      4. metadata-eval 74.2%

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

      5. sub-neg 74.2%

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

      6. *-commutative 74.2%

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

      7. fmm-def 74.2%

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

      8. metadata-eval 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in re around 0 49.3%

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

   9. Taylor expanded in im around inf 49.3%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.25:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.3% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. exp-0 57.2%

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

   3. associate-*l/ 57.2%

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

   4. cos-neg 57.2%

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

   5. associate-*l* 57.2%

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

   6. associate-*r/ 57.2%

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

   7. exp-0 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. *-commutative 57.2%

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

   10. neg-sub0 57.2%

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

   11. cos-neg 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in im around 0 50.0%

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

6. Taylor expanded in re around 0 29.3%

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

7. Final simplification 29.3%

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

Alternative 14: 3.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -256 \end{array} \]

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 -256.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -256.0;
}

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 * (-256.0d0)
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 * -256.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -256.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -256.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -256.0;
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 * -256.0), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. exp-0 57.2%

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

   3. associate-*l/ 57.2%

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

   4. cos-neg 57.2%

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

   5. associate-*l* 57.2%

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

   6. associate-*r/ 57.2%

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

   7. exp-0 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. *-commutative 57.2%

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

   10. neg-sub0 57.2%

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

   11. cos-neg 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in im around 0 50.0%

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

7. Applied egg-rr 2.7%

   \[\leadsto 0.5 \cdot \color{blue}{-512} \]

8. Final simplification 2.7%

   \[\leadsto -256 \]
9. Add Preprocessing

Alternative 15: 3.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -0.0008680555555555555 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -0.0008680555555555555))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -0.0008680555555555555;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-0.0008680555555555555d0)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -0.0008680555555555555;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -0.0008680555555555555

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -0.0008680555555555555)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -0.0008680555555555555;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -0.0008680555555555555), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. exp-0 57.2%

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

   3. associate-*l/ 57.2%

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

   4. cos-neg 57.2%

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

   5. associate-*l* 57.2%

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

   6. associate-*r/ 57.2%

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

   7. exp-0 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. *-commutative 57.2%

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

   10. neg-sub0 57.2%

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

   11. cos-neg 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in im around 0 50.0%

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

7. Applied egg-rr 2.8%

   \[\leadsto 0.5 \cdot \color{blue}{-0.001736111111111111} \]

8. Final simplification 2.8%

   \[\leadsto -0.0008680555555555555 \]
9. Add Preprocessing

Alternative 16: 3.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -2.540263171264543 \cdot 10^{-5} \end{array} \]

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 -2.540263171264543e-5))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -2.540263171264543e-5;
}

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 * (-2.540263171264543d-5)
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 * -2.540263171264543e-5;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -2.540263171264543e-5

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -2.540263171264543e-5)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -2.540263171264543e-5;
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 * -2.540263171264543e-5), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. /-rgt-identity 57.2%

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

   2. exp-0 57.2%

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

   3. associate-*l/ 57.2%

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

   4. cos-neg 57.2%

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

   5. associate-*l* 57.2%

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

   6. associate-*r/ 57.2%

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

   7. exp-0 57.2%

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

   8. /-rgt-identity 57.2%

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

   9. *-commutative 57.2%

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

   10. neg-sub0 57.2%

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

   11. cos-neg 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in im around 0 50.0%

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

7. Applied egg-rr 2.8%

   \[\leadsto 0.5 \cdot \color{blue}{-5.080526342529086 \cdot 10^{-5}} \]

8. Final simplification 2.8%

   \[\leadsto -2.540263171264543 \cdot 10^{-5} \]
9. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024089 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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