math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.6%

Time: 10.3s

Alternatives: 18

Speedup: 14.0×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \log \left(e^{\cos re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= (- (exp (- im_m)) (exp im_m)) -1e+100)
    (* 0.5 (* (- (- 1.0 im_m) (exp im_m)) (log (exp (cos re)))))
    (*
     0.5
     (*
      (cos re)
      (*
       im_m
       (-
        (*
         (pow im_m 2.0)
         (-
          (*
           (pow im_m 2.0)
           (- (* (pow im_m 2.0) -0.0003968253968253968) 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 tmp;
	if ((exp(-im_m) - exp(im_m)) <= -1e+100) {
		tmp = 0.5 * (((1.0 - im_m) - exp(im_m)) * log(exp(cos(re))));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.0003968253968253968) - 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) :: tmp
    if ((exp(-im_m) - exp(im_m)) <= (-1d+100)) then
        tmp = 0.5d0 * (((1.0d0 - im_m) - exp(im_m)) * log(exp(cos(re))))
    else
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.0003968253968253968d0)) - 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 tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -1e+100) {
		tmp = 0.5 * (((1.0 - im_m) - Math.exp(im_m)) * Math.log(Math.exp(Math.cos(re))));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.0003968253968253968) - 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):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -1e+100:
		tmp = 0.5 * (((1.0 - im_m) - math.exp(im_m)) * math.log(math.exp(math.cos(re))))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.0003968253968253968) - 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)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -1e+100)
		tmp = Float64(0.5 * Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * log(exp(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) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 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)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -1e+100)
		tmp = 0.5 * (((1.0 - im_m) - exp(im_m)) * log(exp(cos(re))));
	else
		tmp = 0.5 * (cos(re) * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 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_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -1e+100], N[(0.5 * N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Log[N[Exp[N[Cos[re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $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)) < -1.00000000000000002e100

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.00000000000000002e100 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 95.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(1 - im\right) - e^{im}\right) \cdot \log \left(e^{\cos re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -0.05)
      (* 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 <= -0.05) {
		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 <= (-0.05d0)) 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 <= -0.05) {
		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 <= -0.05:
		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 <= -0.05)
		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 <= -0.05)
		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, -0.05], 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)) < -0.050000000000000003

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

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

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 91.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.016666666666666666 - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + {im\_m}^{2} \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* 0.5 (* t_0 (cos re)))
      (*
       im_m
       (* (cos re) (+ -1.0 (* (pow im_m 2.0) -0.16666666666666666))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = 0.5 * (t_0 * cos(re));
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (pow(im_m, 2.0) * -0.16666666666666666)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.02d0)) then
        tmp = 0.5d0 * (t_0 * cos(re))
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m ** 2.0d0) * (-0.16666666666666666d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = 0.5 * (t_0 * Math.cos(re));
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (Math.pow(im_m, 2.0) * -0.16666666666666666)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.02:
		tmp = 0.5 * (t_0 * math.cos(re))
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (math.pow(im_m, 2.0) * -0.16666666666666666)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(0.5 * Float64(t_0 * cos(re)));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64((im_m ^ 2.0) * -0.16666666666666666))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = 0.5 * (t_0 * cos(re));
	else
		tmp = im_m * (cos(re) * (-1.0 + ((im_m ^ 2.0) * -0.16666666666666666)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(0.5 * N[(t$95$0 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   if -0.0200000000000000004 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 91.9%

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

   6. Taylor expanded in im around 0 83.6%

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

      1. associate-*r* 83.6%

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

      2. distribute-rgt-out 83.6%

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

   8. Simplified 83.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 4.9 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im\_m}\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im\_m}^{5}\right) \cdot -0.008333333333333333\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (cos re)))
    (if (<= im_m 4.9e+59)
      (* 0.5 (- (- 1.0 (exp im_m)) im_m))
      (* (* (cos re) (pow im_m 5.0)) -0.008333333333333333)))))

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 <= 235.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 4.9e+59) {
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	} else {
		tmp = (cos(re) * pow(im_m, 5.0)) * -0.008333333333333333;
	}
	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 <= 235.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 4.9d+59) then
        tmp = 0.5d0 * ((1.0d0 - exp(im_m)) - im_m)
    else
        tmp = (cos(re) * (im_m ** 5.0d0)) * (-0.008333333333333333d0)
    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 <= 235.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 4.9e+59) {
		tmp = 0.5 * ((1.0 - Math.exp(im_m)) - im_m);
	} else {
		tmp = (Math.cos(re) * Math.pow(im_m, 5.0)) * -0.008333333333333333;
	}
	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 <= 235.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 4.9e+59:
		tmp = 0.5 * ((1.0 - math.exp(im_m)) - im_m)
	else:
		tmp = (math.cos(re) * math.pow(im_m, 5.0)) * -0.008333333333333333
	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 <= 235.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 4.9e+59)
		tmp = Float64(0.5 * Float64(Float64(1.0 - exp(im_m)) - im_m));
	else
		tmp = Float64(Float64(cos(re) * (im_m ^ 5.0)) * -0.008333333333333333);
	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 <= 235.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 4.9e+59)
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	else
		tmp = (cos(re) * (im_m ^ 5.0)) * -0.008333333333333333;
	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, 235.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 4.9e+59], N[(0.5 * N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 235 < im < 4.90000000000000007e59

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 81.3%

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

      1. +-commutative 81.3%

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

      2. associate--r+ 81.3%

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

   10. Simplified 81.3%

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

   if 4.90000000000000007e59 < 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 98.1%

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

   6. Taylor expanded in im around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4.9 \cdot 10^{+59}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + {im\_m}^{2} \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(1 - e^{im\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.1)
    (* im_m (* (cos re) (+ -1.0 (* (pow im_m 2.0) -0.16666666666666666))))
    (* 0.5 (* (cos re) (- 1.0 (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.1) {
		tmp = im_m * (cos(re) * (-1.0 + (pow(im_m, 2.0) * -0.16666666666666666)));
	} else {
		tmp = 0.5 * (cos(re) * (1.0 - exp(im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.1d0) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m ** 2.0d0) * (-0.16666666666666666d0))))
    else
        tmp = 0.5d0 * (cos(re) * (1.0d0 - exp(im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.1) {
		tmp = im_m * (Math.cos(re) * (-1.0 + (Math.pow(im_m, 2.0) * -0.16666666666666666)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (1.0 - Math.exp(im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.1:
		tmp = im_m * (math.cos(re) * (-1.0 + (math.pow(im_m, 2.0) * -0.16666666666666666)))
	else:
		tmp = 0.5 * (math.cos(re) * (1.0 - math.exp(im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.1)
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64((im_m ^ 2.0) * -0.16666666666666666))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(1.0 - exp(im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.1)
		tmp = im_m * (cos(re) * (-1.0 + ((im_m ^ 2.0) * -0.16666666666666666)));
	else
		tmp = 0.5 * (cos(re) * (1.0 - exp(im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.10000000000000009

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 91.9%

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

   6. Taylor expanded in im around 0 83.6%

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

      1. associate-*r* 83.6%

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

      2. distribute-rgt-out 83.6%

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

   8. Simplified 83.6%

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

   if 2.10000000000000009 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(1 - e^{im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.25:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(1 - e^{im\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.25)
    (* im_m (- (cos re)))
    (* 0.5 (* (cos re) (- 1.0 (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.25) {
		tmp = im_m * -cos(re);
	} else {
		tmp = 0.5 * (cos(re) * (1.0 - exp(im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.25d0) then
        tmp = im_m * -cos(re)
    else
        tmp = 0.5d0 * (cos(re) * (1.0d0 - exp(im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.25) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = 0.5 * (Math.cos(re) * (1.0 - Math.exp(im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.25:
		tmp = im_m * -math.cos(re)
	else:
		tmp = 0.5 * (math.cos(re) * (1.0 - math.exp(im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.25)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(1.0 - exp(im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.25)
		tmp = im_m * -cos(re);
	else
		tmp = 0.5 * (cos(re) * (1.0 - exp(im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.25], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.25

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 63.8%

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

   6. Taylor expanded in im around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. neg-mul-1 63.8%

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

   8. Simplified 63.8%

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

   if 1.25 < 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 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.25:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(1 - e^{im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -0.1:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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
         (*
          im_m
          (-
           (*
            im_m
            (-
             (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
             0.5))
           2.0))))
   (*
    im_s
    (if (<= (cos re) -0.1)
      (* 0.5 (* t_0 (+ 1.0 (* -0.5 (* re re)))))
      (* 0.5 t_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 = im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0);
	double tmp;
	if (cos(re) <= -0.1) {
		tmp = 0.5 * (t_0 * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = 0.5 * t_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 = im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)
    if (cos(re) <= (-0.1d0)) then
        tmp = 0.5d0 * (t_0 * (1.0d0 + ((-0.5d0) * (re * re))))
    else
        tmp = 0.5d0 * t_0
    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 = im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0);
	double tmp;
	if (Math.cos(re) <= -0.1) {
		tmp = 0.5 * (t_0 * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = 0.5 * t_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 = im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)
	tmp = 0
	if math.cos(re) <= -0.1:
		tmp = 0.5 * (t_0 * (1.0 + (-0.5 * (re * re))))
	else:
		tmp = 0.5 * t_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(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))
	tmp = 0.0
	if (cos(re) <= -0.1)
		tmp = Float64(0.5 * Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	else
		tmp = Float64(0.5 * t_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 = im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0);
	tmp = 0.0;
	if (cos(re) <= -0.1)
		tmp = 0.5 * (t_0 * (1.0 + (-0.5 * (re * re))));
	else
		tmp = 0.5 * t_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[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[Cos[re], $MachinePrecision], -0.1], N[(0.5 * N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.10000000000000001

   1. Initial program 54.9%

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

      1. /-rgt-identity 54.9%

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

      2. exp-0 54.9%

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

      3. associate-*l/ 54.9%

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

      4. cos-neg 54.9%

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

      5. associate-*l* 54.9%

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

      6. associate-*r/ 54.9%

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

      7. exp-0 54.9%

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

      8. /-rgt-identity 54.9%

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

      9. *-commutative 54.9%

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

      10. neg-sub0 54.9%

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

      11. cos-neg 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in im around 0 33.8%

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

      1. neg-mul-1 33.8%

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

      2. unsub-neg 33.8%

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

   7. Simplified 33.8%

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

   8. Taylor expanded in im around 0 67.2%

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

   9. Taylor expanded in re around 0 27.1%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 27.1%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Applied egg-rr 27.1%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   if -0.10000000000000001 < (cos.f64 re)

   1. Initial program 58.0%

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

      1. /-rgt-identity 58.0%

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

      2. exp-0 58.0%

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

      3. associate-*l/ 58.0%

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

      4. cos-neg 58.0%

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

      5. associate-*l* 58.0%

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

      6. associate-*r/ 58.0%

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

      7. exp-0 58.0%

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

      8. /-rgt-identity 58.0%

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

      9. *-commutative 58.0%

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

      10. neg-sub0 58.0%

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

      11. cos-neg 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in im around 0 29.1%

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

      1. neg-mul-1 29.1%

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

      2. unsub-neg 29.1%

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

   7. Simplified 29.1%

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

   8. Taylor expanded in im around 0 60.1%

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

   9. Taylor expanded in re around 0 48.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.1:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im\_m}\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664\right) - 0.5\right) - 2\right)\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (cos re)))
    (if (<= im_m 1.4e+77)
      (* 0.5 (- (- 1.0 (exp im_m)) im_m))
      (*
       0.5
       (*
        (cos re)
        (*
         im_m
         (-
          (* im_m (- (* im_m (* im_m -0.041666666666666664)) 0.5))
          2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.4e+77) {
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * (im_m * -0.041666666666666664)) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 235.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.4d+77) then
        tmp = 0.5d0 * ((1.0d0 - exp(im_m)) - im_m)
    else
        tmp = 0.5d0 * (cos(re) * (im_m * ((im_m * ((im_m * (im_m * (-0.041666666666666664d0))) - 0.5d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.4e+77) {
		tmp = 0.5 * ((1.0 - Math.exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * ((im_m * (im_m * -0.041666666666666664)) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 235.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.4e+77:
		tmp = 0.5 * ((1.0 - math.exp(im_m)) - im_m)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * ((im_m * (im_m * -0.041666666666666664)) - 0.5)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 235.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.4e+77)
		tmp = Float64(0.5 * Float64(Float64(1.0 - exp(im_m)) - im_m));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(im_m * -0.041666666666666664)) - 0.5)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 235.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.4e+77)
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	else
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * (im_m * -0.041666666666666664)) - 0.5)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 235.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+77], N[(0.5 * N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(im$95$m * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 235 < im < 1.4e77

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. associate--r+ 75.0%

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

   10. Simplified 75.0%

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

   if 1.4e77 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(-0.041666666666666664 \cdot im\right)} - 0.5\right) - 2\right)\right) \cdot \cos re\right) \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \color{blue}{\left(im \cdot -0.041666666666666664\right)} - 0.5\right) - 2\right)\right) \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664\right) - 0.5\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im\_m}\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (cos re)))
    (if (<= im_m 1.05e+103)
      (* 0.5 (- (- 1.0 (exp im_m)) im_m))
      (*
       0.5
       (*
        (cos re)
        (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.05e+103) {
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 235.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.05d+103) then
        tmp = 0.5d0 * ((1.0d0 - exp(im_m)) - im_m)
    else
        tmp = 0.5d0 * (cos(re) * (im_m * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.05e+103) {
		tmp = 0.5 * ((1.0 - Math.exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 235.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.05e+103:
		tmp = 0.5 * ((1.0 - math.exp(im_m)) - im_m)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 235.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.05e+103)
		tmp = Float64(0.5 * Float64(Float64(1.0 - exp(im_m)) - im_m));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 235.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.05e+103)
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	else
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 235.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.05e+103], N[(0.5 * N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 235 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. associate--r+ 76.7%

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

   10. Simplified 76.7%

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

   if 1.0500000000000001e103 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 760000:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+61} \lor \neg \left(im\_m \leq 2 \cdot 10^{+122}\right):\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{5} \cdot -0.008333333333333333\\ \end{array} \end{array} \]

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 760000.0)
    (* im_m (- (cos re)))
    (if (or (<= im_m 4.2e+61) (not (<= im_m 2e+122)))
      (*
       0.5
       (*
        (*
         im_m
         (-
          (*
           im_m
           (-
            (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
            0.5))
          2.0))
        (+ 1.0 (* -0.5 (* re re)))))
      (* (pow im_m 5.0) -0.008333333333333333)))))

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 <= 760000.0) {
		tmp = im_m * -cos(re);
	} else if ((im_m <= 4.2e+61) || !(im_m <= 2e+122)) {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = pow(im_m, 5.0) * -0.008333333333333333;
	}
	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 <= 760000.0d0) then
        tmp = im_m * -cos(re)
    else if ((im_m <= 4.2d+61) .or. (.not. (im_m <= 2d+122))) then
        tmp = 0.5d0 * ((im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)) * (1.0d0 + ((-0.5d0) * (re * re))))
    else
        tmp = (im_m ** 5.0d0) * (-0.008333333333333333d0)
    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 <= 760000.0) {
		tmp = im_m * -Math.cos(re);
	} else if ((im_m <= 4.2e+61) || !(im_m <= 2e+122)) {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	} else {
		tmp = Math.pow(im_m, 5.0) * -0.008333333333333333;
	}
	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 <= 760000.0:
		tmp = im_m * -math.cos(re)
	elif (im_m <= 4.2e+61) or not (im_m <= 2e+122):
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))))
	else:
		tmp = math.pow(im_m, 5.0) * -0.008333333333333333
	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 <= 760000.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif ((im_m <= 4.2e+61) || !(im_m <= 2e+122))
		tmp = Float64(0.5 * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	else
		tmp = Float64((im_m ^ 5.0) * -0.008333333333333333);
	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 <= 760000.0)
		tmp = im_m * -cos(re);
	elseif ((im_m <= 4.2e+61) || ~((im_m <= 2e+122)))
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	else
		tmp = (im_m ^ 5.0) * -0.008333333333333333;
	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, 760000.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 4.2e+61], N[Not[LessEqual[im$95$m, 2e+122]], $MachinePrecision]], N[(0.5 * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7.6e5

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 7.6e5 < im < 4.2000000000000002e61 or 2.00000000000000003e122 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 66.2%

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

   9. Taylor expanded in re around 0 59.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 59.2%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Applied egg-rr 59.2%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   if 4.2000000000000002e61 < im < 2.00000000000000003e122

   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%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0 75.0%

      \[\leadsto \color{blue}{{im}^{5}} \cdot -0.008333333333333333 \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+61} \lor \neg \left(im \leq 2 \cdot 10^{+122}\right):\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 440:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 4 \cdot 10^{+61}:\\ \;\;\;\;im\_m \cdot \left(-0.5 \cdot {re}^{2}\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+122}:\\ \;\;\;\;{im\_m}^{5} \cdot -0.008333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

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 440.0)
    (* im_m (- (cos re)))
    (if (<= im_m 4e+61)
      (* im_m (* -0.5 (pow re 2.0)))
      (if (<= im_m 2e+122)
        (* (pow im_m 5.0) -0.008333333333333333)
        (*
         0.5
         (*
          (*
           im_m
           (-
            (*
             im_m
             (-
              (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
              0.5))
            2.0))
          (+ 1.0 (* -0.5 (* re re))))))))))

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 <= 440.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 4e+61) {
		tmp = im_m * (-0.5 * pow(re, 2.0));
	} else if (im_m <= 2e+122) {
		tmp = pow(im_m, 5.0) * -0.008333333333333333;
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 440.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 4d+61) then
        tmp = im_m * ((-0.5d0) * (re ** 2.0d0))
    else if (im_m <= 2d+122) then
        tmp = (im_m ** 5.0d0) * (-0.008333333333333333d0)
    else
        tmp = 0.5d0 * ((im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)) * (1.0d0 + ((-0.5d0) * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 4e+61) {
		tmp = im_m * (-0.5 * Math.pow(re, 2.0));
	} else if (im_m <= 2e+122) {
		tmp = Math.pow(im_m, 5.0) * -0.008333333333333333;
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 440.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 4e+61:
		tmp = im_m * (-0.5 * math.pow(re, 2.0))
	elif im_m <= 2e+122:
		tmp = math.pow(im_m, 5.0) * -0.008333333333333333
	else:
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))))
	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 <= 440.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 4e+61)
		tmp = Float64(im_m * Float64(-0.5 * (re ^ 2.0)));
	elseif (im_m <= 2e+122)
		tmp = Float64((im_m ^ 5.0) * -0.008333333333333333);
	else
		tmp = Float64(0.5 * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 440.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 4e+61)
		tmp = im_m * (-0.5 * (re ^ 2.0));
	elseif (im_m <= 2e+122)
		tmp = (im_m ^ 5.0) * -0.008333333333333333;
	else
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 440.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 4e+61], N[(im$95$m * N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2e+122], N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision], N[(0.5 * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 440

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 440 < im < 3.9999999999999998e61

   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 3.4%

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

   6. Taylor expanded in re around 0 8.2%

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

      1. +-commutative 8.2%

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

      2. *-commutative 8.2%

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

      3. distribute-lft-out 8.2%

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

      4. unpow2 8.2%

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

      5. fma-define 8.2%

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

   8. Simplified 8.2%

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

   9. Taylor expanded in re around inf 7.3%

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

       1. *-commutative 7.3%

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

       2. associate-*l* 7.3%

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

       3. *-commutative 7.3%

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

   11. Simplified 7.3%

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

       1. add-sqr-sqrt 7.3%

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

       2. sqrt-unprod 12.8%

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

       3. swap-sqr 12.8%

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

       4. metadata-eval 12.8%

          \[\leadsto im \cdot \sqrt{\color{blue}{0.25} \cdot \left({re}^{2} \cdot {re}^{2}\right)} \]

       5. metadata-eval 12.8%

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

       6. swap-sqr 12.8%

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

       7. sqrt-unprod 0.2%

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

       8. expm1-log1p-u 0.2%

          \[\leadsto im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-0.5 \cdot {re}^{2}} \cdot \sqrt{-0.5 \cdot {re}^{2}}\right)\right)} \]

       9. add-sqr-sqrt 0.8%

          \[\leadsto im \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-0.5 \cdot {re}^{2}}\right)\right) \]

       10. log1p-define 0.6%

           \[\leadsto im \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]

       11. expm1-undefine 0.6%

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

       12. add-exp-log 19.4%

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

       13. +-commutative 19.4%

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

       14. *-commutative 19.4%

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

       15. fma-define 19.4%

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

   13. Applied egg-rr 19.4%

       \[\leadsto im \cdot \color{blue}{\left(\mathsf{fma}\left({re}^{2}, -0.5, 1\right) - 1\right)} \]
   14. Step-by-step derivation

       1. fma-undefine 19.4%

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

       2. *-commutative 19.4%

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

       3. associate--l+ 19.6%

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

       4. metadata-eval 19.6%

          \[\leadsto im \cdot \left(-0.5 \cdot {re}^{2} + \color{blue}{0}\right) \]

       5. +-rgt-identity 19.6%

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

   15. Simplified 19.6%

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

   if 3.9999999999999998e61 < im < 2.00000000000000003e122

   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%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0 75.0%

      \[\leadsto \color{blue}{{im}^{5}} \cdot -0.008333333333333333 \]

   if 2.00000000000000003e122 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in re around 0 83.9%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 83.9%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Applied egg-rr 83.9%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+61}:\\ \;\;\;\;im \cdot \left(-0.5 \cdot {re}^{2}\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+122}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 3.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im\_m}\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 235.0)
    (* im_m (- (cos re)))
    (if (<= im_m 3.7e+154)
      (* 0.5 (- (- 1.0 (exp im_m)) im_m))
      (* im_m (* (cos re) (+ -1.0 (* im_m -0.25))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 3.7e+154) {
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 235.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 3.7d+154) then
        tmp = 0.5d0 * ((1.0d0 - exp(im_m)) - im_m)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + (im_m * (-0.25d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 3.7e+154) {
		tmp = 0.5 * ((1.0 - Math.exp(im_m)) - im_m);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 235.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 3.7e+154:
		tmp = 0.5 * ((1.0 - math.exp(im_m)) - im_m)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (im_m * -0.25)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 235.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 3.7e+154)
		tmp = Float64(0.5 * Float64(Float64(1.0 - exp(im_m)) - im_m));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(im_m * -0.25))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 235.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 3.7e+154)
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	else
		tmp = im_m * (cos(re) * (-1.0 + (im_m * -0.25)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 235.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 3.7e+154], N[(0.5 * N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 235 < im < 3.69999999999999994e154

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. associate--r+ 72.2%

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

   10. Simplified 72.2%

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

   if 3.69999999999999994e154 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 235:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im\_m}\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

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 235.0)
    (* im_m (- (cos re)))
    (if (<= im_m 2.8e+123)
      (* 0.5 (- (- 1.0 (exp im_m)) im_m))
      (*
       0.5
       (*
        (*
         im_m
         (-
          (*
           im_m
           (-
            (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
            0.5))
          2.0))
        (+ 1.0 (* -0.5 (* re re)))))))))

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 <= 235.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 2.8e+123) {
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 235.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 2.8d+123) then
        tmp = 0.5d0 * ((1.0d0 - exp(im_m)) - im_m)
    else
        tmp = 0.5d0 * ((im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)) * (1.0d0 + ((-0.5d0) * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 235.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 2.8e+123) {
		tmp = 0.5 * ((1.0 - Math.exp(im_m)) - im_m);
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 235.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 2.8e+123:
		tmp = 0.5 * ((1.0 - math.exp(im_m)) - im_m)
	else:
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))))
	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 <= 235.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 2.8e+123)
		tmp = Float64(0.5 * Float64(Float64(1.0 - exp(im_m)) - im_m));
	else
		tmp = Float64(0.5 * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 235.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 2.8e+123)
		tmp = 0.5 * ((1.0 - exp(im_m)) - im_m);
	else
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 235.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.8e+123], N[(0.5 * N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 235

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 235 < im < 2.80000000000000011e123

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. associate--r+ 75.8%

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

   10. Simplified 75.8%

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

   if 2.80000000000000011e123 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

   9. Taylor expanded in re around 0 83.9%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 83.9%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Applied egg-rr 83.9%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 235:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \left(\left(1 - e^{im}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2000000:\\ \;\;\;\;im\_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

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 2000000.0)
    (* im_m (- (cos re)))
    (*
     0.5
     (*
      (*
       im_m
       (-
        (*
         im_m
         (-
          (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
          0.5))
        2.0))
      (+ 1.0 (* -0.5 (* re re))))))))

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 <= 2000000.0) {
		tmp = im_m * -cos(re);
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2000000.0d0) then
        tmp = im_m * -cos(re)
    else
        tmp = 0.5d0 * ((im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0)) * (1.0d0 + ((-0.5d0) * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2000000.0) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2000000.0:
		tmp = im_m * -math.cos(re)
	else:
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))))
	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 <= 2000000.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64(0.5 * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * Float64(1.0 + Float64(-0.5 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2000000.0)
		tmp = im_m * -cos(re);
	else
		tmp = 0.5 * ((im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)) * (1.0 + (-0.5 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2000000.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(0.5 * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2e6

   1. Initial program 43.0%

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

      1. /-rgt-identity 43.0%

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

      2. exp-0 43.0%

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

      3. associate-*l/ 43.0%

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

      4. cos-neg 43.0%

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

      5. associate-*l* 43.0%

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

      6. associate-*r/ 43.0%

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

      7. exp-0 43.0%

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

      8. /-rgt-identity 43.0%

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

      9. *-commutative 43.0%

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

      10. neg-sub0 43.0%

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

      11. cos-neg 43.0%

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

   3. Simplified 43.0%

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

   5. Taylor expanded in im around 0 63.5%

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

   6. Taylor expanded in im around 0 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   8. Simplified 63.5%

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

   if 2e6 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 64.7%

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

   9. Taylor expanded in re around 0 55.6%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   10. Step-by-step derivation

       1. unpow2 55.6%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   11. Applied egg-rr 55.6%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 (<= re 7.8e+89)
    (*
     0.5
     (*
      im_m
      (-
       (*
        im_m
        (-
         (* im_m (- (* im_m -0.041666666666666664) 0.16666666666666666))
         0.5))
       2.0)))
    (* im_m (* 0.5 (* re re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 7.8e+89) {
		tmp = 0.5 * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 7.8d+89) then
        tmp = 0.5d0 * (im_m * ((im_m * ((im_m * ((im_m * (-0.041666666666666664d0)) - 0.16666666666666666d0)) - 0.5d0)) - 2.0d0))
    else
        tmp = im_m * (0.5d0 * (re * re))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 7.8e+89) {
		tmp = 0.5 * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 7.8e+89:
		tmp = 0.5 * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0))
	else:
		tmp = im_m * (0.5 * (re * re))
	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 (re <= 7.8e+89)
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0)));
	else
		tmp = Float64(im_m * Float64(0.5 * Float64(re * re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 7.8e+89)
		tmp = 0.5 * (im_m * ((im_m * ((im_m * ((im_m * -0.041666666666666664) - 0.16666666666666666)) - 0.5)) - 2.0));
	else
		tmp = im_m * (0.5 * (re * re));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 7.8e+89], N[(0.5 * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 7.80000000000000021e89

   1. Initial program 58.8%

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

      1. /-rgt-identity 58.8%

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

      2. exp-0 58.8%

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

      3. associate-*l/ 58.8%

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

      4. cos-neg 58.8%

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

      5. associate-*l* 58.8%

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

      6. associate-*r/ 58.8%

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

      7. exp-0 58.8%

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

      8. /-rgt-identity 58.8%

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

      9. *-commutative 58.8%

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

      10. neg-sub0 58.8%

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

      11. cos-neg 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in im around 0 30.6%

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

      1. neg-mul-1 30.6%

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

      2. unsub-neg 30.6%

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

   7. Simplified 30.6%

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

   8. Taylor expanded in im around 0 60.4%

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

   9. Taylor expanded in re around 0 45.2%

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

   if 7.80000000000000021e89 < re

   1. Initial program 49.1%

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

      1. /-rgt-identity 49.1%

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

      2. exp-0 49.1%

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

      3. associate-*l/ 49.1%

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

      4. cos-neg 49.1%

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

      5. associate-*l* 49.1%

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

      6. associate-*r/ 49.1%

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

      7. exp-0 49.1%

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

      8. /-rgt-identity 49.1%

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

      9. *-commutative 49.1%

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

      10. neg-sub0 49.1%

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

      11. cos-neg 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in im around 0 56.6%

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

   6. Taylor expanded in re around 0 15.9%

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

      1. +-commutative 15.9%

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

      2. *-commutative 15.9%

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

      3. distribute-lft-out 15.9%

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

      4. unpow2 15.9%

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

      5. fma-define 15.9%

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

   8. Simplified 15.9%

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

   9. Taylor expanded in re around inf 15.9%

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

       1. *-commutative 15.9%

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

       2. associate-*l* 15.9%

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

       3. *-commutative 15.9%

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

   11. Simplified 15.9%

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

       1. unpow2 27.5%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   13. Applied egg-rr 15.9%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.8 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.041666666666666664 - 0.16666666666666666\right) - 0.5\right) - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.5% accurate, 25.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2000000000:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 2000000000.0) (- im_m) (* im_m (* 0.5 (* re re))))))

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 <= 2000000000.0) {
		tmp = -im_m;
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2000000000.0d0) then
        tmp = -im_m
    else
        tmp = im_m * (0.5d0 * (re * re))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2000000000.0) {
		tmp = -im_m;
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2000000000.0:
		tmp = -im_m
	else:
		tmp = im_m * (0.5 * (re * re))
	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 <= 2000000000.0)
		tmp = Float64(-im_m);
	else
		tmp = Float64(im_m * Float64(0.5 * Float64(re * re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2000000000.0)
		tmp = -im_m;
	else
		tmp = im_m * (0.5 * (re * re));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2000000000.0], (-im$95$m), N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2e9

   1. Initial program 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. exp-0 43.6%

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

      3. associate-*l/ 43.6%

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

      4. cos-neg 43.6%

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

      5. associate-*l* 43.6%

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

      6. associate-*r/ 43.6%

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

      7. exp-0 43.6%

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

      8. /-rgt-identity 43.6%

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

      9. *-commutative 43.6%

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

      10. neg-sub0 43.6%

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

      11. cos-neg 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in im around 0 62.9%

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

   6. Taylor expanded in re around 0 35.8%

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

      1. neg-mul-1 35.8%

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

   8. Simplified 35.8%

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

   if 2e9 < 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 4.9%

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

   6. Taylor expanded in re around 0 19.3%

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

      1. +-commutative 19.3%

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

      2. *-commutative 19.3%

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

      3. distribute-lft-out 19.3%

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

      4. unpow2 19.3%

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

      5. fma-define 19.3%

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

   8. Simplified 19.3%

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

   9. Taylor expanded in re around inf 17.4%

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

       1. *-commutative 17.4%

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

       2. associate-*l* 17.4%

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

       3. *-commutative 17.4%

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

   11. Simplified 17.4%

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

       1. unpow2 57.3%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \left(im \cdot \left(-0.041666666666666664 \cdot im - 0.16666666666666666\right) - 0.5\right) - 2\right)\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   13. Applied egg-rr 17.4%

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

Alternative 17: 29.8% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. /-rgt-identity 57.3%

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

   2. exp-0 57.3%

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

   3. associate-*l/ 57.3%

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

   4. cos-neg 57.3%

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

   5. associate-*l* 57.3%

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

   6. associate-*r/ 57.3%

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

   7. exp-0 57.3%

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

   8. /-rgt-identity 57.3%

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

   9. *-commutative 57.3%

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

   10. neg-sub0 57.3%

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

   11. cos-neg 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in im around 0 48.9%

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

6. Taylor expanded in re around 0 28.0%

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

   1. neg-mul-1 28.0%

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

8. Simplified 28.0%

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

Alternative 18: 3.6% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. /-rgt-identity 57.3%

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

   2. exp-0 57.3%

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

   3. associate-*l/ 57.3%

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

   4. cos-neg 57.3%

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

   5. associate-*l* 57.3%

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

   6. associate-*r/ 57.3%

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

   7. exp-0 57.3%

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

   8. /-rgt-identity 57.3%

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

   9. *-commutative 57.3%

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

   10. neg-sub0 57.3%

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

   11. cos-neg 57.3%

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

3. Simplified 57.3%

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

6. Applied egg-rr 2.6%

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

7. Taylor expanded in re around 0 2.7%

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

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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