math.sin on complex, imaginary part

Percentage Accurate: 54.9% → 99.5%

Time: 11.5s

Alternatives: 14

Speedup: 2.9×

• 50.5% 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: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 - e^{im_m} \leq -\infty:\\ \;\;\;\;\left(e^{im_m} - t_0\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{5} \cdot -0.008333333333333333 + {im_m}^{7} \cdot -0.0001984126984126984\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) (- INFINITY))
      (* (- (exp im_m) t_0) (* (cos re) -0.5))
      (+
       (*
        (cos re)
        (+
         (* (pow im_m 5.0) -0.008333333333333333)
         (* (pow im_m 7.0) -0.0001984126984126984)))
       (* (cos re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -((double) INFINITY)) {
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	} else {
		tmp = (cos(re) * ((pow(im_m, 5.0) * -0.008333333333333333) + (pow(im_m, 7.0) * -0.0001984126984126984))) + (cos(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m));
	}
	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 t_0 = Math.exp(-im_m);
	double tmp;
	if ((t_0 - Math.exp(im_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.exp(im_m) - t_0) * (Math.cos(re) * -0.5);
	} else {
		tmp = (Math.cos(re) * ((Math.pow(im_m, 5.0) * -0.008333333333333333) + (Math.pow(im_m, 7.0) * -0.0001984126984126984))) + (Math.cos(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m)
	tmp = 0
	if (t_0 - math.exp(im_m)) <= -math.inf:
		tmp = (math.exp(im_m) - t_0) * (math.cos(re) * -0.5)
	else:
		tmp = (math.cos(re) * ((math.pow(im_m, 5.0) * -0.008333333333333333) + (math.pow(im_m, 7.0) * -0.0001984126984126984))) + (math.cos(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= Float64(-Inf))
		tmp = Float64(Float64(exp(im_m) - t_0) * Float64(cos(re) * -0.5));
	else
		tmp = Float64(Float64(cos(re) * Float64(Float64((im_m ^ 5.0) * -0.008333333333333333) + Float64((im_m ^ 7.0) * -0.0001984126984126984))) + Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m);
	tmp = 0.0;
	if ((t_0 - exp(im_m)) <= -Inf)
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	else
		tmp = (cos(re) * (((im_m ^ 5.0) * -0.008333333333333333) + ((im_m ^ 7.0) * -0.0001984126984126984))) + (cos(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Exp[im$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sub0-neg 43.0%

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

      3. +-commutative 43.0%

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

      4. neg-sub0 43.0%

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

      5. associate-+l- 43.0%

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

      6. remove-double-neg 43.0%

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

      7. sub0-neg 43.0%

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

      8. neg-sub0 43.0%

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

      9. distribute-rgt-neg-in 43.0%

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

      10. distribute-lft-neg-in 43.0%

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

      11. *-commutative 43.0%

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

      12. sub0-neg 43.0%

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

      13. remove-double-neg 43.0%

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

      14. *-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in im around 0 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 93.9%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

      2. +-commutative 93.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      3. associate-*r* 93.9%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      4. +-rgt-identity 93.9%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} \cdot \cos re + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      5. +-inverses 93.9%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3} + \color{blue}{\left(re - re\right)}\right) \cdot \cos re + -1 \cdot \left(im \cdot \cos re\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      6. associate-*r* 93.9%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(re - re\right)\right) \cdot \cos re + \color{blue}{\left(-1 \cdot im\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      7. distribute-rgt-in 93.9%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + \left(re - re\right)\right) + -1 \cdot im\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      8. +-commutative 93.9%

         \[\leadsto \cos re \cdot \color{blue}{\left(-1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{3} + \left(re - re\right)\right)\right)} + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      9. mul-1-neg 93.9%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(-im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} + \left(re - re\right)\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      10. +-inverses 93.9%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + \left(-0.16666666666666666 \cdot {im}^{3} + \color{blue}{0}\right)\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      11. +-rgt-identity 93.9%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + \color{blue}{-0.16666666666666666 \cdot {im}^{3}}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

      12. *-commutative 93.9%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) + \left(-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right) + -0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)\right) \]

   6. Simplified 93.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right) + \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   7. Taylor expanded in re around inf 93.9%

      \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) \]
   8. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \cos re} + \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) \]

   9. Simplified 93.9%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \cos re} + \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot -0.5\\ t_1 := e^{-im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_1 - e^{im_m} \leq -\infty:\\ \;\;\;\;\left(e^{im_m} - t_1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left({im_m}^{7} \cdot 0.0003968253968253968 + \left({im_m}^{5} \cdot 0.016666666666666666 + \left({im_m}^{3} \cdot 0.3333333333333333 + im_m \cdot 2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) -0.5)) (t_1 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_1 (exp im_m)) (- INFINITY))
      (* (- (exp im_m) t_1) t_0)
      (*
       t_0
       (+
        (* (pow im_m 7.0) 0.0003968253968253968)
        (+
         (* (pow im_m 5.0) 0.016666666666666666)
         (+ (* (pow im_m 3.0) 0.3333333333333333) (* im_m 2.0)))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = cos(re) * -0.5;
	double t_1 = exp(-im_m);
	double tmp;
	if ((t_1 - exp(im_m)) <= -((double) INFINITY)) {
		tmp = (exp(im_m) - t_1) * t_0;
	} else {
		tmp = t_0 * ((pow(im_m, 7.0) * 0.0003968253968253968) + ((pow(im_m, 5.0) * 0.016666666666666666) + ((pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 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 t_0 = Math.cos(re) * -0.5;
	double t_1 = Math.exp(-im_m);
	double tmp;
	if ((t_1 - Math.exp(im_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.exp(im_m) - t_1) * t_0;
	} else {
		tmp = t_0 * ((Math.pow(im_m, 7.0) * 0.0003968253968253968) + ((Math.pow(im_m, 5.0) * 0.016666666666666666) + ((Math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0))));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.cos(re) * -0.5
	t_1 = math.exp(-im_m)
	tmp = 0
	if (t_1 - math.exp(im_m)) <= -math.inf:
		tmp = (math.exp(im_m) - t_1) * t_0
	else:
		tmp = t_0 * ((math.pow(im_m, 7.0) * 0.0003968253968253968) + ((math.pow(im_m, 5.0) * 0.016666666666666666) + ((math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0))))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(cos(re) * -0.5)
	t_1 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_1 - exp(im_m)) <= Float64(-Inf))
		tmp = Float64(Float64(exp(im_m) - t_1) * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64((im_m ^ 7.0) * 0.0003968253968253968) + Float64(Float64((im_m ^ 5.0) * 0.016666666666666666) + Float64(Float64((im_m ^ 3.0) * 0.3333333333333333) + Float64(im_m * 2.0)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = cos(re) * -0.5;
	t_1 = exp(-im_m);
	tmp = 0.0;
	if ((t_1 - exp(im_m)) <= -Inf)
		tmp = (exp(im_m) - t_1) * t_0;
	else
		tmp = t_0 * (((im_m ^ 7.0) * 0.0003968253968253968) + (((im_m ^ 5.0) * 0.016666666666666666) + (((im_m ^ 3.0) * 0.3333333333333333) + (im_m * 2.0))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$1 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Exp[im$95$m], $MachinePrecision] - t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(N[Power[im$95$m, 7.0], $MachinePrecision] * 0.0003968253968253968), $MachinePrecision] + N[(N[(N[Power[im$95$m, 5.0], $MachinePrecision] * 0.016666666666666666), $MachinePrecision] + N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sub0-neg 43.0%

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

      3. +-commutative 43.0%

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

      4. neg-sub0 43.0%

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

      5. associate-+l- 43.0%

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

      6. remove-double-neg 43.0%

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

      7. sub0-neg 43.0%

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

      8. neg-sub0 43.0%

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

      9. distribute-rgt-neg-in 43.0%

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

      10. distribute-lft-neg-in 43.0%

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

      11. *-commutative 43.0%

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

      12. sub0-neg 43.0%

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

      13. remove-double-neg 43.0%

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

      14. *-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in im around 0 93.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot {im}^{7} + \left(0.016666666666666666 \cdot {im}^{5} + \left(0.3333333333333333 \cdot {im}^{3} + 2 \cdot im\right)\right)\right)} \cdot \left(\cos re \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot -0.5\right) \cdot \left({im}^{7} \cdot 0.0003968253968253968 + \left({im}^{5} \cdot 0.016666666666666666 + \left({im}^{3} \cdot 0.3333333333333333 + im \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 - e^{im_m} \leq -0.04:\\ \;\;\;\;\left(e^{im_m} - t_0\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{5} \cdot -0.008333333333333333 + -0.16666666666666666 \cdot {im_m}^{3}\right) - im_m \cdot \cos re\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.04)
      (* (- (exp im_m) t_0) (* (cos re) -0.5))
      (-
       (*
        (cos re)
        (+
         (* (pow im_m 5.0) -0.008333333333333333)
         (* -0.16666666666666666 (pow im_m 3.0))))
       (* im_m (cos re)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.04) {
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	} else {
		tmp = (cos(re) * ((pow(im_m, 5.0) * -0.008333333333333333) + (-0.16666666666666666 * pow(im_m, 3.0)))) - (im_m * cos(re));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m)
    if ((t_0 - exp(im_m)) <= (-0.04d0)) then
        tmp = (exp(im_m) - t_0) * (cos(re) * (-0.5d0))
    else
        tmp = (cos(re) * (((im_m ** 5.0d0) * (-0.008333333333333333d0)) + ((-0.16666666666666666d0) * (im_m ** 3.0d0)))) - (im_m * cos(re))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m);
	double tmp;
	if ((t_0 - Math.exp(im_m)) <= -0.04) {
		tmp = (Math.exp(im_m) - t_0) * (Math.cos(re) * -0.5);
	} else {
		tmp = (Math.cos(re) * ((Math.pow(im_m, 5.0) * -0.008333333333333333) + (-0.16666666666666666 * Math.pow(im_m, 3.0)))) - (im_m * Math.cos(re));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m)
	tmp = 0
	if (t_0 - math.exp(im_m)) <= -0.04:
		tmp = (math.exp(im_m) - t_0) * (math.cos(re) * -0.5)
	else:
		tmp = (math.cos(re) * ((math.pow(im_m, 5.0) * -0.008333333333333333) + (-0.16666666666666666 * math.pow(im_m, 3.0)))) - (im_m * math.cos(re))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.04)
		tmp = Float64(Float64(exp(im_m) - t_0) * Float64(cos(re) * -0.5));
	else
		tmp = Float64(Float64(cos(re) * Float64(Float64((im_m ^ 5.0) * -0.008333333333333333) + Float64(-0.16666666666666666 * (im_m ^ 3.0)))) - Float64(im_m * cos(re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m);
	tmp = 0.0;
	if ((t_0 - exp(im_m)) <= -0.04)
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	else
		tmp = (cos(re) * (((im_m ^ 5.0) * -0.008333333333333333) + (-0.16666666666666666 * (im_m ^ 3.0)))) - (im_m * cos(re));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[Exp[im$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0400000000000000008

   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. sub-neg 99.9%

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

      2. sub0-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub0-neg 99.9%

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

      8. neg-sub0 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. *-commutative 99.9%

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

      12. sub0-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -0.0400000000000000008 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 42.7%

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

      1. sub-neg 42.7%

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

      2. sub0-neg 42.7%

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

      3. +-commutative 42.7%

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

      4. neg-sub0 42.7%

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

      5. associate-+l- 42.7%

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

      6. remove-double-neg 42.7%

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

      7. sub0-neg 42.7%

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

      8. neg-sub0 42.7%

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

      9. distribute-rgt-neg-in 42.7%

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

      10. distribute-lft-neg-in 42.7%

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

      11. *-commutative 42.7%

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

      12. sub0-neg 42.7%

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

      13. remove-double-neg 42.7%

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

      14. *-commutative 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in im around 0 93.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 93.3%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 93.3%

         \[\leadsto \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 93.3%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) - im \cdot \cos re} \]

      4. associate-*r* 93.3%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)\right) - im \cdot \cos re \]

      5. associate-*r* 93.3%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re}\right) - im \cdot \cos re \]

      6. distribute-rgt-out 93.3%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + -0.008333333333333333 \cdot {im}^{5}\right)} - im \cdot \cos re \]

      7. *-commutative 93.3%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} + -0.008333333333333333 \cdot {im}^{5}\right) - im \cdot \cos re \]

      8. *-commutative 93.3%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \color{blue}{{im}^{5} \cdot -0.008333333333333333}\right) - im \cdot \cos re \]

   6. Simplified 93.3%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \cos re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.04:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + -0.16666666666666666 \cdot {im}^{3}\right) - im \cdot \cos re\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m}\\ t_1 := \cos re \cdot -0.5\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 - e^{im_m} \leq -0.04:\\ \;\;\;\;\left(e^{im_m} - t_0\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left({im_m}^{5} \cdot 0.016666666666666666 + \left({im_m}^{3} \cdot 0.3333333333333333 + im_m \cdot 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))) (t_1 (* (cos re) -0.5)))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.04)
      (* (- (exp im_m) t_0) t_1)
      (*
       t_1
       (+
        (* (pow im_m 5.0) 0.016666666666666666)
        (+ (* (pow im_m 3.0) 0.3333333333333333) (* im_m 2.0))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double t_1 = cos(re) * -0.5;
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.04) {
		tmp = (exp(im_m) - t_0) * t_1;
	} else {
		tmp = t_1 * ((pow(im_m, 5.0) * 0.016666666666666666) + ((pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0)));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m)
    t_1 = cos(re) * (-0.5d0)
    if ((t_0 - exp(im_m)) <= (-0.04d0)) then
        tmp = (exp(im_m) - t_0) * t_1
    else
        tmp = t_1 * (((im_m ** 5.0d0) * 0.016666666666666666d0) + (((im_m ** 3.0d0) * 0.3333333333333333d0) + (im_m * 2.0d0)))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m);
	double t_1 = Math.cos(re) * -0.5;
	double tmp;
	if ((t_0 - Math.exp(im_m)) <= -0.04) {
		tmp = (Math.exp(im_m) - t_0) * t_1;
	} else {
		tmp = t_1 * ((Math.pow(im_m, 5.0) * 0.016666666666666666) + ((Math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0)));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m)
	t_1 = math.cos(re) * -0.5
	tmp = 0
	if (t_0 - math.exp(im_m)) <= -0.04:
		tmp = (math.exp(im_m) - t_0) * t_1
	else:
		tmp = t_1 * ((math.pow(im_m, 5.0) * 0.016666666666666666) + ((math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0)))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	t_1 = Float64(cos(re) * -0.5)
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.04)
		tmp = Float64(Float64(exp(im_m) - t_0) * t_1);
	else
		tmp = Float64(t_1 * Float64(Float64((im_m ^ 5.0) * 0.016666666666666666) + Float64(Float64((im_m ^ 3.0) * 0.3333333333333333) + Float64(im_m * 2.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m);
	t_1 = cos(re) * -0.5;
	tmp = 0.0;
	if ((t_0 - exp(im_m)) <= -0.04)
		tmp = (exp(im_m) - t_0) * t_1;
	else
		tmp = t_1 * (((im_m ^ 5.0) * 0.016666666666666666) + (((im_m ^ 3.0) * 0.3333333333333333) + (im_m * 2.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * -0.5), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(N[Exp[im$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[(N[Power[im$95$m, 5.0], $MachinePrecision] * 0.016666666666666666), $MachinePrecision] + N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0400000000000000008

   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. sub-neg 99.9%

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

      2. sub0-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub0-neg 99.9%

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

      8. neg-sub0 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. *-commutative 99.9%

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

      12. sub0-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -0.0400000000000000008 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 42.7%

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

      1. sub-neg 42.7%

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

      2. sub0-neg 42.7%

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

      3. +-commutative 42.7%

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

      4. neg-sub0 42.7%

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

      5. associate-+l- 42.7%

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

      6. remove-double-neg 42.7%

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

      7. sub0-neg 42.7%

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

      8. neg-sub0 42.7%

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

      9. distribute-rgt-neg-in 42.7%

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

      10. distribute-lft-neg-in 42.7%

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

      11. *-commutative 42.7%

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

      12. sub0-neg 42.7%

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

      13. remove-double-neg 42.7%

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

      14. *-commutative 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in im around 0 93.3%

      \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot {im}^{5} + \left(0.3333333333333333 \cdot {im}^{3} + 2 \cdot im\right)\right)} \cdot \left(\cos re \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.04:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot -0.5\right) \cdot \left({im}^{5} \cdot 0.016666666666666666 + \left({im}^{3} \cdot 0.3333333333333333 + im \cdot 2\right)\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 - e^{im_m} \leq -0.005:\\ \;\;\;\;\left(e^{im_m} - t_0\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -0.005)
      (* (- (exp im_m) t_0) (* (cos re) -0.5))
      (* (cos re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m);
	double tmp;
	if ((t_0 - exp(im_m)) <= -0.005) {
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m)
    if ((t_0 - exp(im_m)) <= (-0.005d0)) then
        tmp = (exp(im_m) - t_0) * (cos(re) * (-0.5d0))
    else
        tmp = cos(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m);
	double tmp;
	if ((t_0 - Math.exp(im_m)) <= -0.005) {
		tmp = (Math.exp(im_m) - t_0) * (Math.cos(re) * -0.5);
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m)
	tmp = 0
	if (t_0 - math.exp(im_m)) <= -0.005:
		tmp = (math.exp(im_m) - t_0) * (math.cos(re) * -0.5)
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = exp(Float64(-im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -0.005)
		tmp = Float64(Float64(exp(im_m) - t_0) * Float64(cos(re) * -0.5));
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m);
	tmp = 0.0;
	if ((t_0 - exp(im_m)) <= -0.005)
		tmp = (exp(im_m) - t_0) * (cos(re) * -0.5);
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[Exp[im$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0050000000000000001

   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. sub-neg 99.9%

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

      2. sub0-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub0-neg 99.9%

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

      8. neg-sub0 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. distribute-lft-neg-in 99.9%

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

      11. *-commutative 99.9%

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

      12. sub0-neg 99.9%

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

      13. remove-double-neg 99.9%

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

      14. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 42.7%

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

      1. sub-neg 42.7%

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

      2. sub0-neg 42.7%

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

      3. +-commutative 42.7%

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

      4. neg-sub0 42.7%

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

      5. associate-+l- 42.7%

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

      6. remove-double-neg 42.7%

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

      7. sub0-neg 42.7%

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

      8. neg-sub0 42.7%

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

      9. distribute-rgt-neg-in 42.7%

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

      10. distribute-lft-neg-in 42.7%

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

      11. *-commutative 42.7%

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

      12. sub0-neg 42.7%

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

      13. remove-double-neg 42.7%

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

      14. *-commutative 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in im around 0 89.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 89.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 89.7%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 89.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 89.7%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 89.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 89.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 89.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.005:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot \left(\cos re \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 6: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.025:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im_m \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{im_m} - e^{-im_m}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{7} \cdot -0.0001984126984126984\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.025)
    (* im_m (- (cos re)))
    (if (<= im_m 1.06e+44)
      (* (- (exp im_m) (exp (- im_m))) -0.5)
      (* (cos re) (* (pow im_m 7.0) -0.0001984126984126984))))))

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 <= 0.025) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.06e+44) {
		tmp = (exp(im_m) - exp(-im_m)) * -0.5;
	} else {
		tmp = cos(re) * (pow(im_m, 7.0) * -0.0001984126984126984);
	}
	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 <= 0.025d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.06d+44) then
        tmp = (exp(im_m) - exp(-im_m)) * (-0.5d0)
    else
        tmp = cos(re) * ((im_m ** 7.0d0) * (-0.0001984126984126984d0))
    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 <= 0.025) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.06e+44) {
		tmp = (Math.exp(im_m) - Math.exp(-im_m)) * -0.5;
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 7.0) * -0.0001984126984126984);
	}
	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 <= 0.025:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.06e+44:
		tmp = (math.exp(im_m) - math.exp(-im_m)) * -0.5
	else:
		tmp = math.cos(re) * (math.pow(im_m, 7.0) * -0.0001984126984126984)
	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 <= 0.025)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.06e+44)
		tmp = Float64(Float64(exp(im_m) - exp(Float64(-im_m))) * -0.5);
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 7.0) * -0.0001984126984126984));
	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 <= 0.025)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.06e+44)
		tmp = (exp(im_m) - exp(-im_m)) * -0.5;
	else
		tmp = cos(re) * ((im_m ^ 7.0) * -0.0001984126984126984);
	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, 0.025], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.06e+44], N[(N[(N[Exp[im$95$m], $MachinePrecision] - N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.025000000000000001

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sub0-neg 43.0%

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

      3. +-commutative 43.0%

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

      4. neg-sub0 43.0%

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

      5. associate-+l- 43.0%

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

      6. remove-double-neg 43.0%

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

      7. sub0-neg 43.0%

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

      8. neg-sub0 43.0%

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

      9. distribute-rgt-neg-in 43.0%

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

      10. distribute-lft-neg-in 43.0%

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

      11. *-commutative 43.0%

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

      12. sub0-neg 43.0%

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

      13. remove-double-neg 43.0%

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

      14. *-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. mul-1-neg 63.1%

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

   6. Simplified 63.1%

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

   if 0.025000000000000001 < im < 1.06e44

   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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 62.5%

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

   if 1.06e44 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot {im}^{7} + \left(0.016666666666666666 \cdot {im}^{5} + \left(0.3333333333333333 \cdot {im}^{3} + 2 \cdot im\right)\right)\right)} \cdot \left(\cos re \cdot -0.5\right) \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.095:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \mathbf{elif}\;im_m \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{im_m} - e^{-im_m}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{7} \cdot -0.0001984126984126984\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.095)
    (* (cos re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 1.06e+44)
      (* (- (exp im_m) (exp (- im_m))) -0.5)
      (* (cos re) (* (pow im_m 7.0) -0.0001984126984126984))))))

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 <= 0.095) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 1.06e+44) {
		tmp = (exp(im_m) - exp(-im_m)) * -0.5;
	} else {
		tmp = cos(re) * (pow(im_m, 7.0) * -0.0001984126984126984);
	}
	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 <= 0.095d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 1.06d+44) then
        tmp = (exp(im_m) - exp(-im_m)) * (-0.5d0)
    else
        tmp = cos(re) * ((im_m ** 7.0d0) * (-0.0001984126984126984d0))
    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 <= 0.095) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 1.06e+44) {
		tmp = (Math.exp(im_m) - Math.exp(-im_m)) * -0.5;
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 7.0) * -0.0001984126984126984);
	}
	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 <= 0.095:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 1.06e+44:
		tmp = (math.exp(im_m) - math.exp(-im_m)) * -0.5
	else:
		tmp = math.cos(re) * (math.pow(im_m, 7.0) * -0.0001984126984126984)
	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 <= 0.095)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 1.06e+44)
		tmp = Float64(Float64(exp(im_m) - exp(Float64(-im_m))) * -0.5);
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 7.0) * -0.0001984126984126984));
	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 <= 0.095)
		tmp = cos(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 1.06e+44)
		tmp = (exp(im_m) - exp(-im_m)) * -0.5;
	else
		tmp = cos(re) * ((im_m ^ 7.0) * -0.0001984126984126984);
	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, 0.095], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.06e+44], N[(N[(N[Exp[im$95$m], $MachinePrecision] - N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.095000000000000001

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sub0-neg 43.0%

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

      3. +-commutative 43.0%

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

      4. neg-sub0 43.0%

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

      5. associate-+l- 43.0%

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

      6. remove-double-neg 43.0%

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

      7. sub0-neg 43.0%

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

      8. neg-sub0 43.0%

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

      9. distribute-rgt-neg-in 43.0%

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

      10. distribute-lft-neg-in 43.0%

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

      11. *-commutative 43.0%

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

      12. sub0-neg 43.0%

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

      13. remove-double-neg 43.0%

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

      14. *-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in im around 0 89.6%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 89.6%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 89.6%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 89.6%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 89.6%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 89.6%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 89.6%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if 0.095000000000000001 < im < 1.06e44

   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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 62.5%

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

   if 1.06e44 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot {im}^{7} + \left(0.016666666666666666 \cdot {im}^{5} + \left(0.3333333333333333 \cdot {im}^{3} + 2 \cdot im\right)\right)\right)} \cdot \left(\cos re \cdot -0.5\right) \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot {im}^{7}\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.095:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 8: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.025:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im_m} - e^{-im_m}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.025)
    (* im_m (- (cos re)))
    (* (- (exp im_m) (exp (- 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 <= 0.025) {
		tmp = im_m * -cos(re);
	} else {
		tmp = (exp(im_m) - exp(-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 <= 0.025d0) then
        tmp = im_m * -cos(re)
    else
        tmp = (exp(im_m) - exp(-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 <= 0.025) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = (Math.exp(im_m) - Math.exp(-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 <= 0.025:
		tmp = im_m * -math.cos(re)
	else:
		tmp = (math.exp(im_m) - math.exp(-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 <= 0.025)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64(Float64(exp(im_m) - exp(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 <= 0.025)
		tmp = im_m * -cos(re);
	else
		tmp = (exp(im_m) - exp(-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, 0.025], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[(N[Exp[im$95$m], $MachinePrecision] - N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.025000000000000001

   1. Initial program 43.0%

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

      1. sub-neg 43.0%

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

      2. sub0-neg 43.0%

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

      3. +-commutative 43.0%

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

      4. neg-sub0 43.0%

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

      5. associate-+l- 43.0%

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

      6. remove-double-neg 43.0%

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

      7. sub0-neg 43.0%

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

      8. neg-sub0 43.0%

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

      9. distribute-rgt-neg-in 43.0%

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

      10. distribute-lft-neg-in 43.0%

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

      11. *-commutative 43.0%

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

      12. sub0-neg 43.0%

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

      13. remove-double-neg 43.0%

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

      14. *-commutative 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. mul-1-neg 63.1%

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

   6. Simplified 63.1%

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

   if 0.025000000000000001 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 67.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.025:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} - e^{-im}\right) \cdot -0.5\\ \end{array} \]

Alternative 9: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 44000:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im_m \leq 1.55 \cdot 10^{+117}:\\ \;\;\;\;im_m \cdot \left(-1 - -0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left({im_m}^{3} \cdot 0.3333333333333333 + im_m \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 44000.0)
    (* im_m (- (cos re)))
    (if (<= im_m 1.55e+117)
      (* im_m (- -1.0 (* -0.5 (pow re 2.0))))
      (* -0.5 (+ (* (pow im_m 3.0) 0.3333333333333333) (* im_m 2.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 44000.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.55e+117) {
		tmp = im_m * (-1.0 - (-0.5 * pow(re, 2.0)));
	} else {
		tmp = -0.5 * ((pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 44000.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.55d+117) then
        tmp = im_m * ((-1.0d0) - ((-0.5d0) * (re ** 2.0d0)))
    else
        tmp = (-0.5d0) * (((im_m ** 3.0d0) * 0.3333333333333333d0) + (im_m * 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 44000.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.55e+117) {
		tmp = im_m * (-1.0 - (-0.5 * Math.pow(re, 2.0)));
	} else {
		tmp = -0.5 * ((Math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 44000.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.55e+117:
		tmp = im_m * (-1.0 - (-0.5 * math.pow(re, 2.0)))
	else:
		tmp = -0.5 * ((math.pow(im_m, 3.0) * 0.3333333333333333) + (im_m * 2.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 44000.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.55e+117)
		tmp = Float64(im_m * Float64(-1.0 - Float64(-0.5 * (re ^ 2.0))));
	else
		tmp = Float64(-0.5 * Float64(Float64((im_m ^ 3.0) * 0.3333333333333333) + Float64(im_m * 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 44000.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.55e+117)
		tmp = im_m * (-1.0 - (-0.5 * (re ^ 2.0)));
	else
		tmp = -0.5 * (((im_m ^ 3.0) * 0.3333333333333333) + (im_m * 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 44000.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.55e+117], N[(im$95$m * N[(-1.0 - N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 44000

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sub0-neg 43.3%

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

      3. +-commutative 43.3%

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

      4. neg-sub0 43.3%

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

      5. associate-+l- 43.3%

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

      6. remove-double-neg 43.3%

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

      7. sub0-neg 43.3%

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

      8. neg-sub0 43.3%

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

      9. distribute-rgt-neg-in 43.3%

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

      10. distribute-lft-neg-in 43.3%

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

      11. *-commutative 43.3%

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

      12. sub0-neg 43.3%

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

      13. remove-double-neg 43.3%

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

      14. *-commutative 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in im around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. mul-1-neg 62.8%

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

   6. Simplified 62.8%

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

   if 44000 < im < 1.54999999999999988e117

   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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.7%

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

      1. associate-*r* 3.7%

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

      2. mul-1-neg 3.7%

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

   6. Simplified 3.7%

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

   7. Taylor expanded in re around 0 37.4%

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

   if 1.54999999999999988e117 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 74.4%

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

   5. Taylor expanded in im around 0 74.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 44000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+117}:\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left({im}^{3} \cdot 0.3333333333333333 + im \cdot 2\right)\\ \end{array} \]

Alternative 10: 59.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 21500:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im_m \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;{re}^{2} \cdot \left(im_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(-0.5 \cdot {re}^{2} + -1\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 21500.0)
    (* im_m (- (cos re)))
    (if (<= im_m 1.55e+195)
      (* (pow re 2.0) (* im_m 0.5))
      (* im_m (+ (* -0.5 (pow re 2.0)) -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 <= 21500.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 1.55e+195) {
		tmp = pow(re, 2.0) * (im_m * 0.5);
	} else {
		tmp = im_m * ((-0.5 * pow(re, 2.0)) + -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 <= 21500.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 1.55d+195) then
        tmp = (re ** 2.0d0) * (im_m * 0.5d0)
    else
        tmp = im_m * (((-0.5d0) * (re ** 2.0d0)) + (-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 <= 21500.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 1.55e+195) {
		tmp = Math.pow(re, 2.0) * (im_m * 0.5);
	} else {
		tmp = im_m * ((-0.5 * Math.pow(re, 2.0)) + -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 <= 21500.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 1.55e+195:
		tmp = math.pow(re, 2.0) * (im_m * 0.5)
	else:
		tmp = im_m * ((-0.5 * math.pow(re, 2.0)) + -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 <= 21500.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 1.55e+195)
		tmp = Float64((re ^ 2.0) * Float64(im_m * 0.5));
	else
		tmp = Float64(im_m * Float64(Float64(-0.5 * (re ^ 2.0)) + -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 <= 21500.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 1.55e+195)
		tmp = (re ^ 2.0) * (im_m * 0.5);
	else
		tmp = im_m * ((-0.5 * (re ^ 2.0)) + -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, 21500.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.55e+195], N[(N[Power[re, 2.0], $MachinePrecision] * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 21500

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sub0-neg 43.3%

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

      3. +-commutative 43.3%

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

      4. neg-sub0 43.3%

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

      5. associate-+l- 43.3%

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

      6. remove-double-neg 43.3%

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

      7. sub0-neg 43.3%

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

      8. neg-sub0 43.3%

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

      9. distribute-rgt-neg-in 43.3%

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

      10. distribute-lft-neg-in 43.3%

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

      11. *-commutative 43.3%

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

      12. sub0-neg 43.3%

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

      13. remove-double-neg 43.3%

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

      14. *-commutative 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in im around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. mul-1-neg 62.8%

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

   6. Simplified 62.8%

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

   if 21500 < im < 1.5500000000000001e195

   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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.1%

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

      1. associate-*r* 4.1%

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

      2. mul-1-neg 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in re around 0 27.3%

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

   8. Taylor expanded in re around inf 25.7%

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

      1. associate-*r* 25.7%

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

      2. *-commutative 25.7%

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

      3. *-commutative 25.7%

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

   10. Simplified 25.7%

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

   if 1.5500000000000001e195 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 7.4%

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

      1. associate-*r* 7.4%

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

      2. mul-1-neg 7.4%

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

   6. Simplified 7.4%

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

   7. Taylor expanded in re around 0 31.3%

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

      1. distribute-lft-in 31.3%

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

      2. *-rgt-identity 31.3%

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

      3. neg-sub0 31.3%

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

      4. associate-+l- 31.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 20.8%

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

      9. add-sqr-sqrt 20.8%

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

   9. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{0 - \left(im - im \cdot \left(-0.5 \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--r- 20.8%

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

       2. sub0-neg 20.8%

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

       3. mul-1-neg 20.8%

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

       4. *-commutative 20.8%

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

       5. distribute-lft-out 20.8%

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

   11. Simplified 20.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 21500:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+195}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-0.5 \cdot {re}^{2} + -1\right)\\ \end{array} \]

Alternative 11: 59.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.5 \cdot {re}^{2}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 10500:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im_m \leq 9.2 \cdot 10^{+195}:\\ \;\;\;\;im_m \cdot \left(-1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(t_0 + -1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* -0.5 (pow re 2.0))))
   (*
    im_s
    (if (<= im_m 10500.0)
      (* im_m (- (cos re)))
      (if (<= im_m 9.2e+195) (* im_m (- -1.0 t_0)) (* im_m (+ t_0 -1.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.5 * pow(re, 2.0);
	double tmp;
	if (im_m <= 10500.0) {
		tmp = im_m * -cos(re);
	} else if (im_m <= 9.2e+195) {
		tmp = im_m * (-1.0 - t_0);
	} else {
		tmp = im_m * (t_0 + -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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (re ** 2.0d0)
    if (im_m <= 10500.0d0) then
        tmp = im_m * -cos(re)
    else if (im_m <= 9.2d+195) then
        tmp = im_m * ((-1.0d0) - t_0)
    else
        tmp = im_m * (t_0 + (-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 t_0 = -0.5 * Math.pow(re, 2.0);
	double tmp;
	if (im_m <= 10500.0) {
		tmp = im_m * -Math.cos(re);
	} else if (im_m <= 9.2e+195) {
		tmp = im_m * (-1.0 - t_0);
	} else {
		tmp = im_m * (t_0 + -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):
	t_0 = -0.5 * math.pow(re, 2.0)
	tmp = 0
	if im_m <= 10500.0:
		tmp = im_m * -math.cos(re)
	elif im_m <= 9.2e+195:
		tmp = im_m * (-1.0 - t_0)
	else:
		tmp = im_m * (t_0 + -1.0)
	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(-0.5 * (re ^ 2.0))
	tmp = 0.0
	if (im_m <= 10500.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	elseif (im_m <= 9.2e+195)
		tmp = Float64(im_m * Float64(-1.0 - t_0));
	else
		tmp = Float64(im_m * Float64(t_0 + -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)
	t_0 = -0.5 * (re ^ 2.0);
	tmp = 0.0;
	if (im_m <= 10500.0)
		tmp = im_m * -cos(re);
	elseif (im_m <= 9.2e+195)
		tmp = im_m * (-1.0 - t_0);
	else
		tmp = im_m * (t_0 + -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_] := Block[{t$95$0 = N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 10500.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 9.2e+195], N[(im$95$m * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 10500

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sub0-neg 43.3%

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

      3. +-commutative 43.3%

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

      4. neg-sub0 43.3%

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

      5. associate-+l- 43.3%

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

      6. remove-double-neg 43.3%

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

      7. sub0-neg 43.3%

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

      8. neg-sub0 43.3%

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

      9. distribute-rgt-neg-in 43.3%

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

      10. distribute-lft-neg-in 43.3%

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

      11. *-commutative 43.3%

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

      12. sub0-neg 43.3%

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

      13. remove-double-neg 43.3%

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

      14. *-commutative 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in im around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. mul-1-neg 62.8%

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

   6. Simplified 62.8%

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

   if 10500 < im < 9.2000000000000005e195

   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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.1%

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

      1. associate-*r* 4.1%

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

      2. mul-1-neg 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in re around 0 27.3%

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

   if 9.2000000000000005e195 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 7.4%

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

      1. associate-*r* 7.4%

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

      2. mul-1-neg 7.4%

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

   6. Simplified 7.4%

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

   7. Taylor expanded in re around 0 31.3%

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

      1. distribute-lft-in 31.3%

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

      2. *-rgt-identity 31.3%

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

      3. neg-sub0 31.3%

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

      4. associate-+l- 31.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 37.9%

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

      7. sqr-neg 37.9%

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

      8. sqrt-unprod 20.8%

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

      9. add-sqr-sqrt 20.8%

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

   9. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{0 - \left(im - im \cdot \left(-0.5 \cdot {re}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. associate--r- 20.8%

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

       2. sub0-neg 20.8%

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

       3. mul-1-neg 20.8%

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

       4. *-commutative 20.8%

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

       5. distribute-lft-out 20.8%

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

   11. Simplified 20.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+195}:\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot {re}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-0.5 \cdot {re}^{2} + -1\right)\\ \end{array} \]

Alternative 12: 58.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 10500:\\ \;\;\;\;im_m \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(im_m \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 10500.0) (* im_m (- (cos re))) (* (pow re 2.0) (* im_m 0.5)))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 10500.0) {
		tmp = im_m * -cos(re);
	} else {
		tmp = pow(re, 2.0) * (im_m * 0.5);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 10500.0d0) then
        tmp = im_m * -cos(re)
    else
        tmp = (re ** 2.0d0) * (im_m * 0.5d0)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 10500.0) {
		tmp = im_m * -Math.cos(re);
	} else {
		tmp = Math.pow(re, 2.0) * (im_m * 0.5);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 10500.0:
		tmp = im_m * -math.cos(re)
	else:
		tmp = math.pow(re, 2.0) * (im_m * 0.5)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 10500.0)
		tmp = Float64(im_m * Float64(-cos(re)));
	else
		tmp = Float64((re ^ 2.0) * Float64(im_m * 0.5));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 10500.0)
		tmp = im_m * -cos(re);
	else
		tmp = (re ^ 2.0) * (im_m * 0.5);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 10500.0], N[(im$95$m * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[Power[re, 2.0], $MachinePrecision] * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 10500

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sub0-neg 43.3%

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

      3. +-commutative 43.3%

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

      4. neg-sub0 43.3%

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

      5. associate-+l- 43.3%

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

      6. remove-double-neg 43.3%

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

      7. sub0-neg 43.3%

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

      8. neg-sub0 43.3%

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

      9. distribute-rgt-neg-in 43.3%

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

      10. distribute-lft-neg-in 43.3%

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

      11. *-commutative 43.3%

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

      12. sub0-neg 43.3%

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

      13. remove-double-neg 43.3%

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

      14. *-commutative 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in im around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. mul-1-neg 62.8%

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

   6. Simplified 62.8%

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

   if 10500 < 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. sub-neg 100.0%

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

      2. sub0-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub0-neg 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-lft-neg-in 100.0%

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

      11. *-commutative 100.0%

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

      12. sub0-neg 100.0%

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

      13. remove-double-neg 100.0%

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

      14. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.5%

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

      1. associate-*r* 5.5%

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

      2. mul-1-neg 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in re around 0 29.0%

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

   8. Taylor expanded in re around inf 26.9%

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

      1. associate-*r* 26.9%

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

      2. *-commutative 26.9%

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

      3. *-commutative 26.9%

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

   10. Simplified 26.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10500:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 51.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \left(im_m \cdot \left(-\cos re\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m (- (cos re)))))

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 * -cos(re));
}

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 * -cos(re))
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 * -Math.cos(re));
}

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 * -math.cos(re))

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 * Float64(-cos(re))))
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 * -cos(re));
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   2. sub0-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. neg-sub0 57.9%

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

   5. associate-+l- 57.9%

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

   6. remove-double-neg 57.9%

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

   7. sub0-neg 57.9%

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

   8. neg-sub0 57.9%

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

   9. distribute-rgt-neg-in 57.9%

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

   10. distribute-lft-neg-in 57.9%

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

   11. *-commutative 57.9%

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

   12. sub0-neg 57.9%

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

   13. remove-double-neg 57.9%

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

   14. *-commutative 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in im around 0 48.0%

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

   1. associate-*r* 48.0%

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

   2. mul-1-neg 48.0%

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

6. Simplified 48.0%

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

7. Final simplification 48.0%

   \[\leadsto im \cdot \left(-\cos re\right) \]

Alternative 14: 29.0% 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 1 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 57.9%

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

   1. sub-neg 57.9%

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

   2. sub0-neg 57.9%

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

   3. +-commutative 57.9%

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

   4. neg-sub0 57.9%

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

   5. associate-+l- 57.9%

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

   6. remove-double-neg 57.9%

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

   7. sub0-neg 57.9%

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

   8. neg-sub0 57.9%

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

   9. distribute-rgt-neg-in 57.9%

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

   10. distribute-lft-neg-in 57.9%

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

   11. *-commutative 57.9%

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

   12. sub0-neg 57.9%

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

   13. remove-double-neg 57.9%

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

   14. *-commutative 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in im around 0 48.0%

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

   1. associate-*r* 48.0%

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

   2. mul-1-neg 48.0%

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

6. Simplified 48.0%

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

7. Taylor expanded in re around 0 24.6%

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

   1. mul-1-neg 24.6%

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

9. Simplified 24.6%

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

10. Final simplification 24.6%

    \[\leadsto -im \]

Developer target: 99.8% accurate, 0.8× 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 2023326 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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