math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.4%

Time: 8.7s

Alternatives: 11

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im\_m}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \end{array} \]

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)) (- INFINITY))
    (* 0.5 (* (- 27.0 (exp im_m)) (cos re)))
    (*
     0.5
     (*
      (cos re)
      (*
       im_m
       (-
        (*
         (* im_m im_m)
         (-
          (*
           (* im_m im_m)
           (- (* (* im_m im_m) -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)) <= -((double) INFINITY)) {
		tmp = 0.5 * ((27.0 - exp(im_m)) * cos(re));
	} else {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = 0.5 * ((27.0 - Math.exp(im_m)) * Math.cos(re));
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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)) <= -math.inf:
		tmp = 0.5 * ((27.0 - math.exp(im_m)) * math.cos(re))
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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)) <= Float64(-Inf))
		tmp = Float64(0.5 * Float64(Float64(27.0 - exp(im_m)) * cos(re)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -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)) <= -Inf)
		tmp = 0.5 * ((27.0 - exp(im_m)) * cos(re));
	else
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -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], (-Infinity)], N[(0.5 * N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $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)) < -inf.0

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 96.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 96.3%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   7. Applied egg-rr 96.3%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   8. Step-by-step derivation

      1. unpow2 96.3%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   9. Applied egg-rr 96.3%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   10. Step-by-step derivation

       1. unpow2 96.3%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   11. Applied egg-rr 96.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(\left(27 - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.73 \lor \neg \left(im\_m \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 0.73) (not (<= im_m 3.35e+44)))
    (*
     0.5
     (*
      (cos re)
      (*
       im_m
       (-
        (*
         (* im_m im_m)
         (-
          (*
           (* im_m im_m)
           (- (* (* im_m im_m) -0.0003968253968253968) 0.016666666666666666))
          0.3333333333333333))
        2.0))))
    (* 0.5 (- (exp (- im_m)) (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.73) || !(im_m <= 3.35e+44)) {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 0.73d0) .or. (.not. (im_m <= 3.35d+44))) then
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    else
        tmp = 0.5d0 * (exp(-im_m) - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 0.73) || !(im_m <= 3.35e+44)) {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (Math.exp(-im_m) - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 0.73) or not (im_m <= 3.35e+44):
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)))
	else:
		tmp = 0.5 * (math.exp(-im_m) - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 0.73) || !(im_m <= 3.35e+44))
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 0.73) || ~((im_m <= 3.35e+44)))
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	else
		tmp = 0.5 * (exp(-im_m) - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 0.73], N[Not[LessEqual[im$95$m, 3.35e+44]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.72999999999999998 or 3.35000000000000018e44 < im

   1. Initial program 55.9%

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

      1. /-rgt-identity 55.9%

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

      2. exp-0 55.9%

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

      3. associate-*l/ 55.9%

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

      4. cos-neg 55.9%

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

      5. associate-*l* 55.9%

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

      6. associate-*r/ 55.9%

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

      7. exp-0 55.9%

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

      8. /-rgt-identity 55.9%

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

      9. *-commutative 55.9%

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

      10. neg-sub0 55.9%

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

      11. cos-neg 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in im around 0 97.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   7. Applied egg-rr 97.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   8. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   9. Applied egg-rr 97.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   10. Step-by-step derivation

       1. unpow2 97.2%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   11. Applied egg-rr 97.2%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   if 0.72999999999999998 < im < 3.35000000000000018e44

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 70.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.73 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.9 \lor \neg \left(im\_m \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 6.9) (not (<= im_m 3.35e+44)))
    (*
     0.5
     (*
      (cos re)
      (*
       im_m
       (-
        (*
         (* im_m im_m)
         (-
          (*
           (* im_m im_m)
           (- (* (* im_m im_m) -0.0003968253968253968) 0.016666666666666666))
          0.3333333333333333))
        2.0))))
    (* 0.5 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 6.9) || !(im_m <= 3.35e+44)) {
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((im_m <= 6.9d0) .or. (.not. (im_m <= 3.35d+44))) then
        tmp = 0.5d0 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0)))
    else
        tmp = 0.5d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((im_m <= 6.9) || !(im_m <= 3.35e+44)) {
		tmp = 0.5 * (Math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	} else {
		tmp = 0.5 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (im_m <= 6.9) or not (im_m <= 3.35e+44):
		tmp = 0.5 * (math.cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)))
	else:
		tmp = 0.5 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if ((im_m <= 6.9) || !(im_m <= 3.35e+44))
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * Float64(Float64(Float64(im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))));
	else
		tmp = Float64(0.5 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((im_m <= 6.9) || ~((im_m <= 3.35e+44)))
		tmp = 0.5 * (cos(re) * (im_m * (((im_m * im_m) * (((im_m * im_m) * (((im_m * im_m) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	else
		tmp = 0.5 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[im$95$m, 6.9], N[Not[LessEqual[im$95$m, 3.35e+44]], $MachinePrecision]], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.9000000000000004 or 3.35000000000000018e44 < im

   1. Initial program 55.9%

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

      1. /-rgt-identity 55.9%

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

      2. exp-0 55.9%

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

      3. associate-*l/ 55.9%

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

      4. cos-neg 55.9%

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

      5. associate-*l* 55.9%

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

      6. associate-*r/ 55.9%

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

      7. exp-0 55.9%

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

      8. /-rgt-identity 55.9%

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

      9. *-commutative 55.9%

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

      10. neg-sub0 55.9%

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

      11. cos-neg 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in im around 0 97.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   7. Applied egg-rr 97.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   8. Step-by-step derivation

      1. unpow2 97.2%

         \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   9. Applied egg-rr 97.2%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]
   10. Step-by-step derivation

       1. unpow2 97.2%

          \[\leadsto 0.5 \cdot \left(\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot \color{blue}{\left(im \cdot im\right)} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   11. Applied egg-rr 97.2%

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\left(im \cdot im\right) \cdot \left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right) \cdot \cos re\right) \]

   if 6.9000000000000004 < im < 3.35000000000000018e44

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 70.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.9 \lor \neg \left(im \leq 3.35 \cdot 10^{+44}\right):\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.2:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) + -1\right)\right)\right)\\ \end{array} \end{array} \]

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 5.2)
    (* (cos re) (- im_m))
    (if (<= im_m 1.05e+103)
      (* 0.5 (- 27.0 (exp im_m)))
      (*
       0.5
       (*
        (cos re)
        (+
         26.0
         (*
          im_m
          (+ (* im_m (- (* im_m -0.16666666666666666) 0.5)) -1.0)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.2) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 1.05e+103) {
		tmp = 0.5 * (27.0 - exp(im_m));
	} else {
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) + -1.0))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5.2d0) then
        tmp = cos(re) * -im_m
    else if (im_m <= 1.05d+103) then
        tmp = 0.5d0 * (27.0d0 - exp(im_m))
    else
        tmp = 0.5d0 * (cos(re) * (26.0d0 + (im_m * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) + (-1.0d0)))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.2) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 1.05e+103) {
		tmp = 0.5 * (27.0 - Math.exp(im_m));
	} else {
		tmp = 0.5 * (Math.cos(re) * (26.0 + (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) + -1.0))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.2:
		tmp = math.cos(re) * -im_m
	elif im_m <= 1.05e+103:
		tmp = 0.5 * (27.0 - math.exp(im_m))
	else:
		tmp = 0.5 * (math.cos(re) * (26.0 + (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) + -1.0))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.2)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 1.05e+103)
		tmp = Float64(0.5 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(26.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) + -1.0)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5.2)
		tmp = cos(re) * -im_m;
	elseif (im_m <= 1.05e+103)
		tmp = 0.5 * (27.0 - exp(im_m));
	else
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) + -1.0))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.2], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 1.05e+103], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 64.6%

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

   6. Taylor expanded in im around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. mul-1-neg 64.6%

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

   8. Simplified 64.6%

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

   if 5.20000000000000018 < 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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 70.4%

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

   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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(26 + im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 1\right)\right)} \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) + -1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im\_m \cdot \left(im\_m \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.7)
    (* (cos re) (- im_m))
    (if (<= im_m 1.85e+154)
      (* 0.5 (- 27.0 (exp im_m)))
      (* 0.5 (* (cos re) (+ 26.0 (* im_m (* im_m -0.5)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 1.85e+154) {
		tmp = 0.5 * (27.0 - exp(im_m));
	} else {
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.7d0) then
        tmp = cos(re) * -im_m
    else if (im_m <= 1.85d+154) then
        tmp = 0.5d0 * (27.0d0 - exp(im_m))
    else
        tmp = 0.5d0 * (cos(re) * (26.0d0 + (im_m * (im_m * (-0.5d0)))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 1.85e+154) {
		tmp = 0.5 * (27.0 - Math.exp(im_m));
	} else {
		tmp = 0.5 * (Math.cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.7:
		tmp = math.cos(re) * -im_m
	elif im_m <= 1.85e+154:
		tmp = 0.5 * (27.0 - math.exp(im_m))
	else:
		tmp = 0.5 * (math.cos(re) * (26.0 + (im_m * (im_m * -0.5))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.7)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 1.85e+154)
		tmp = Float64(0.5 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(26.0 + Float64(im_m * Float64(im_m * -0.5)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.7)
		tmp = cos(re) * -im_m;
	elseif (im_m <= 1.85e+154)
		tmp = 0.5 * (27.0 - exp(im_m));
	else
		tmp = 0.5 * (cos(re) * (26.0 + (im_m * (im_m * -0.5))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.7], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 1.85e+154], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(26.0 + N[(im$95$m * N[(im$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.7000000000000002

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 64.6%

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

   6. Taylor expanded in im around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. mul-1-neg 64.6%

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

   8. Simplified 64.6%

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

   if 3.7000000000000002 < im < 1.84999999999999997e154

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 76.3%

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

   if 1.84999999999999997e154 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(26 + im \cdot \left(im \cdot -0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.6:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 4.6) (* (cos re) (- im_m)) (* 0.5 (- 27.0 (exp im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = 0.5 * (27.0 - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.6d0) then
        tmp = cos(re) * -im_m
    else
        tmp = 0.5d0 * (27.0d0 - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.6) {
		tmp = Math.cos(re) * -im_m;
	} else {
		tmp = 0.5 * (27.0 - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.6:
		tmp = math.cos(re) * -im_m
	else:
		tmp = 0.5 * (27.0 - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.6)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(0.5 * Float64(27.0 - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.6)
		tmp = cos(re) * -im_m;
	else
		tmp = 0.5 * (27.0 - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.6], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(0.5 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.5999999999999996

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 64.6%

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

   6. Taylor expanded in im around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. mul-1-neg 64.6%

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

   8. Simplified 64.6%

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

   if 4.5999999999999996 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0 73.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.6:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(27 - e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 26 + im\_m \cdot \left(im\_m \cdot -0.5 + -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 320000:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+226}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 (+ 26.0 (* im_m (+ (* im_m -0.5) -1.0)))))
   (*
    im_s
    (if (<= im_m 320000.0)
      (* (cos re) (- im_m))
      (if (<= im_m 2e+226)
        (* 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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	double tmp;
	if (im_m <= 320000.0) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 2e+226) {
		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 = 26.0d0 + (im_m * ((im_m * (-0.5d0)) + (-1.0d0)))
    if (im_m <= 320000.0d0) then
        tmp = cos(re) * -im_m
    else if (im_m <= 2d+226) 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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	double tmp;
	if (im_m <= 320000.0) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 2e+226) {
		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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0))
	tmp = 0
	if im_m <= 320000.0:
		tmp = math.cos(re) * -im_m
	elif im_m <= 2e+226:
		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(26.0 + Float64(im_m * Float64(Float64(im_m * -0.5) + -1.0)))
	tmp = 0.0
	if (im_m <= 320000.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 2e+226)
		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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	tmp = 0.0;
	if (im_m <= 320000.0)
		tmp = cos(re) * -im_m;
	elseif (im_m <= 2e+226)
		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[(26.0 + N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 320000.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 2e+226], 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 3 regimes

2. if im < 3.2e5

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 64.6%

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

   6. Taylor expanded in im around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. mul-1-neg 64.6%

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

   8. Simplified 64.6%

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

   if 3.2e5 < im < 1.99999999999999992e226

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 35.6%

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

   8. Taylor expanded in re around 0 41.5%

      \[\leadsto 0.5 \cdot \left(\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 41.5%

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

   10. Applied egg-rr 41.5%

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

   if 1.99999999999999992e226 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   8. Taylor expanded in re around 0 69.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 320000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+226}:\\ \;\;\;\;0.5 \cdot \left(\left(26 + im \cdot \left(im \cdot -0.5 + -1\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im \cdot \left(im \cdot -0.5 + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 10.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 26 + im\_m \cdot \left(im\_m \cdot -0.5 + -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 470000:\\ \;\;\;\;-im\_m\\ \mathbf{elif}\;im\_m \leq 4 \cdot 10^{+225}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 (+ 26.0 (* im_m (+ (* im_m -0.5) -1.0)))))
   (*
    im_s
    (if (<= im_m 470000.0)
      (- im_m)
      (if (<= im_m 4e+225)
        (* 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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	double tmp;
	if (im_m <= 470000.0) {
		tmp = -im_m;
	} else if (im_m <= 4e+225) {
		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 = 26.0d0 + (im_m * ((im_m * (-0.5d0)) + (-1.0d0)))
    if (im_m <= 470000.0d0) then
        tmp = -im_m
    else if (im_m <= 4d+225) 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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	double tmp;
	if (im_m <= 470000.0) {
		tmp = -im_m;
	} else if (im_m <= 4e+225) {
		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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0))
	tmp = 0
	if im_m <= 470000.0:
		tmp = -im_m
	elif im_m <= 4e+225:
		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(26.0 + Float64(im_m * Float64(Float64(im_m * -0.5) + -1.0)))
	tmp = 0.0
	if (im_m <= 470000.0)
		tmp = Float64(-im_m);
	elseif (im_m <= 4e+225)
		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 = 26.0 + (im_m * ((im_m * -0.5) + -1.0));
	tmp = 0.0;
	if (im_m <= 470000.0)
		tmp = -im_m;
	elseif (im_m <= 4e+225)
		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[(26.0 + N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 470000.0], (-im$95$m), If[LessEqual[im$95$m, 4e+225], 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 3 regimes

2. if im < 4.7e5

   1. Initial program 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. exp-0 41.9%

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

      3. associate-*l/ 41.9%

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

      4. cos-neg 41.9%

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

      5. associate-*l* 41.9%

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

      6. associate-*r/ 41.9%

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

      7. exp-0 41.9%

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

      8. /-rgt-identity 41.9%

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

      9. *-commutative 41.9%

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

      10. neg-sub0 41.9%

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

      11. cos-neg 41.9%

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

   3. Simplified 41.9%

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

   5. Taylor expanded in im around 0 64.6%

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

   6. Taylor expanded in re around 0 38.5%

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

      1. mul-1-neg 38.5%

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

   8. Simplified 38.5%

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

   if 4.7e5 < im < 3.99999999999999971e225

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 35.6%

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

   8. Taylor expanded in re around 0 41.5%

      \[\leadsto 0.5 \cdot \left(\left(26 + im \cdot \left(-0.5 \cdot im - 1\right)\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 41.5%

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

   10. Applied egg-rr 41.5%

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

   if 3.99999999999999971e225 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   8. Taylor expanded in re around 0 69.2%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 470000:\\ \;\;\;\;-im\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+225}:\\ \;\;\;\;0.5 \cdot \left(\left(26 + im \cdot \left(im \cdot -0.5 + -1\right)\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im \cdot \left(im \cdot -0.5 + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.9% accurate, 19.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0185:\\ \;\;\;\;-im\_m\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im\_m \cdot \left(im\_m \cdot -0.5 + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.0185)
    (- im_m)
    (* 0.5 (+ 26.0 (* im_m (+ (* im_m -0.5) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0185) {
		tmp = -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * ((im_m * -0.5) + -1.0)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.0185d0) then
        tmp = -im_m
    else
        tmp = 0.5d0 * (26.0d0 + (im_m * ((im_m * (-0.5d0)) + (-1.0d0))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.0185) {
		tmp = -im_m;
	} else {
		tmp = 0.5 * (26.0 + (im_m * ((im_m * -0.5) + -1.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.0185:
		tmp = -im_m
	else:
		tmp = 0.5 * (26.0 + (im_m * ((im_m * -0.5) + -1.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.0185)
		tmp = Float64(-im_m);
	else
		tmp = Float64(0.5 * Float64(26.0 + Float64(im_m * Float64(Float64(im_m * -0.5) + -1.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.0185)
		tmp = -im_m;
	else
		tmp = 0.5 * (26.0 + (im_m * ((im_m * -0.5) + -1.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.0185], (-im$95$m), N[(0.5 * N[(26.0 + N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.0184999999999999991

   1. Initial program 41.6%

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

      1. /-rgt-identity 41.6%

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

      2. exp-0 41.6%

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

      3. associate-*l/ 41.6%

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

      4. cos-neg 41.6%

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

      5. associate-*l* 41.6%

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

      6. associate-*r/ 41.6%

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

      7. exp-0 41.6%

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

      8. /-rgt-identity 41.6%

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

      9. *-commutative 41.6%

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

      10. neg-sub0 41.6%

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

      11. cos-neg 41.6%

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

   3. Simplified 41.6%

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

   5. Taylor expanded in im around 0 64.7%

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

   6. Taylor expanded in re around 0 38.7%

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

      1. mul-1-neg 38.7%

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

   8. Simplified 38.7%

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

   if 0.0184999999999999991 < im

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

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

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in im around 0 47.1%

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

   8. Taylor expanded in re around 0 33.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0185:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(26 + im \cdot \left(im \cdot -0.5 + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.5% 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 57.6%

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

   1. /-rgt-identity 57.6%

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

   2. exp-0 57.6%

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

   3. associate-*l/ 57.6%

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

   4. cos-neg 57.6%

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

   5. associate-*l* 57.6%

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

   6. associate-*r/ 57.6%

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

   7. exp-0 57.6%

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

   8. /-rgt-identity 57.6%

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

   9. *-commutative 57.6%

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

   10. neg-sub0 57.6%

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

   11. cos-neg 57.6%

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

3. Simplified 57.6%

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

5. Taylor expanded in im around 0 48.6%

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

6. Taylor expanded in re around 0 29.2%

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

   1. mul-1-neg 29.2%

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

8. Simplified 29.2%

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

Alternative 11: 3.6% accurate, 309.0× speedup

Math

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

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 57.6%

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

   1. /-rgt-identity 57.6%

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

   2. exp-0 57.6%

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

   3. associate-*l/ 57.6%

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

   4. cos-neg 57.6%

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

   5. associate-*l* 57.6%

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

   6. associate-*r/ 57.6%

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

   7. exp-0 57.6%

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

   8. /-rgt-identity 57.6%

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

   9. *-commutative 57.6%

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

   10. neg-sub0 57.6%

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

   11. cos-neg 57.6%

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

3. Simplified 57.6%

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

6. Applied egg-rr 2.7%

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

   1. metadata-eval 2.7%

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

8. Applied egg-rr 2.7%

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

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024144 
(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))))