math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.1%

Time: 8.3s

Alternatives: 13

Speedup: 2.8×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(1 - im\_m\right) - e^{im\_m}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -2e+64)
    (* 0.5 (* (- (- 1.0 im_m) (exp im_m)) (cos re)))
    (* (cos re) (- im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+64) {
		tmp = 0.5 * (((1.0 - im_m) - exp(im_m)) * cos(re));
	} else {
		tmp = cos(re) * -im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) - exp(im_m)) <= (-2d+64)) then
        tmp = 0.5d0 * (((1.0d0 - im_m) - exp(im_m)) * cos(re))
    else
        tmp = cos(re) * -im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -2e+64) {
		tmp = 0.5 * (((1.0 - im_m) - Math.exp(im_m)) * Math.cos(re));
	} else {
		tmp = Math.cos(re) * -im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -2e+64:
		tmp = 0.5 * (((1.0 - im_m) - math.exp(im_m)) * math.cos(re))
	else:
		tmp = math.cos(re) * -im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+64)
		tmp = Float64(0.5 * Float64(Float64(Float64(1.0 - im_m) - exp(im_m)) * cos(re)));
	else
		tmp = Float64(cos(re) * Float64(-im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -2e+64)
		tmp = 0.5 * (((1.0 - im_m) - exp(im_m)) * cos(re));
	else
		tmp = cos(re) * -im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+64], N[(0.5 * N[(N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -2.00000000000000004e64

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -2.00000000000000004e64 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 39.4%

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

      1. /-rgt-identity 39.4%

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

      2. exp-0 39.4%

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

      3. associate-*l/ 39.4%

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

      4. cos-neg 39.4%

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

      5. associate-*l* 39.4%

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

      6. associate-*r/ 39.4%

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

      7. exp-0 39.4%

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

      8. /-rgt-identity 39.4%

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

      9. *-commutative 39.4%

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

      10. neg-sub0 39.4%

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

      11. cos-neg 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in im around 0 66.8%

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

      1. add-cube-cbrt 65.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 65.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 65.5%

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

      6. neg-mul-1 65.5%

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

      7. *-commutative 65.5%

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

   7. Applied egg-rr 65.5%

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

      1. rem-cube-cbrt 66.8%

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

      2. distribute-rgt-neg-out 66.8%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.8%

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

   9. Applied egg-rr 66.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(1 - im\right) - e^{im}\right) \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.7:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.7)
    (* (cos re) (- im_m))
    (if (<= im_m 3.1e+102)
      (* 0.5 (- (- (expm1 im_m)) im_m))
      (*
       0.5
       (*
        (cos re)
        (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.7) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 3.1e+102) {
		tmp = 0.5 * (-expm1(im_m) - im_m);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.7) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 3.1e+102) {
		tmp = 0.5 * (-Math.expm1(im_m) - im_m);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.7:
		tmp = math.cos(re) * -im_m
	elif im_m <= 3.1e+102:
		tmp = 0.5 * (-math.expm1(im_m) - im_m)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.7)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 3.1e+102)
		tmp = Float64(0.5 * Float64(Float64(-expm1(im_m)) - im_m));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.7], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 3.1e+102], N[(0.5 * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.7000000000000002

   1. Initial program 39.4%

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

      1. /-rgt-identity 39.4%

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

      2. exp-0 39.4%

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

      3. associate-*l/ 39.4%

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

      4. cos-neg 39.4%

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

      5. associate-*l* 39.4%

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

      6. associate-*r/ 39.4%

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

      7. exp-0 39.4%

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

      8. /-rgt-identity 39.4%

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

      9. *-commutative 39.4%

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

      10. neg-sub0 39.4%

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

      11. cos-neg 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in im around 0 66.8%

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

      1. add-cube-cbrt 65.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 65.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 65.5%

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

      6. neg-mul-1 65.5%

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

      7. *-commutative 65.5%

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

   7. Applied egg-rr 65.5%

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

      1. rem-cube-cbrt 66.8%

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

      2. distribute-rgt-neg-out 66.8%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.8%

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

   9. Applied egg-rr 66.8%

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

   if 2.7000000000000002 < im < 3.09999999999999987e102

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 88.2%

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

      1. associate--r+ 88.2%

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

      2. unsub-neg 88.2%

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

      3. +-commutative 88.2%

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

      4. associate-+r- 88.2%

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

      5. neg-sub0 88.2%

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

      6. associate-+l- 88.2%

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

      7. sub0-neg 88.2%

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

      8. expm1-define 88.2%

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

   10. Simplified 88.2%

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

   if 3.09999999999999987e102 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 98.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.4:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.4)
    (* (cos re) (- im_m))
    (if (<= im_m 1.9e+154)
      (* 0.5 (- (- (expm1 im_m)) im_m))
      (* 0.5 (* (cos re) (* im_m (- (* im_m -0.5) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.4) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 1.9e+154) {
		tmp = 0.5 * (-expm1(im_m) - im_m);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.4) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 1.9e+154) {
		tmp = 0.5 * (-Math.expm1(im_m) - im_m);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.4:
		tmp = math.cos(re) * -im_m
	elif im_m <= 1.9e+154:
		tmp = 0.5 * (-math.expm1(im_m) - im_m)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((im_m * -0.5) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.4)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 1.9e+154)
		tmp = Float64(0.5 * Float64(Float64(-expm1(im_m)) - im_m));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(im_m * -0.5) - 2.0))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.4], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 1.9e+154], N[(0.5 * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) - im$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.39999999999999991

   1. Initial program 39.4%

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

      1. /-rgt-identity 39.4%

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

      2. exp-0 39.4%

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

      3. associate-*l/ 39.4%

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

      4. cos-neg 39.4%

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

      5. associate-*l* 39.4%

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

      6. associate-*r/ 39.4%

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

      7. exp-0 39.4%

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

      8. /-rgt-identity 39.4%

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

      9. *-commutative 39.4%

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

      10. neg-sub0 39.4%

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

      11. cos-neg 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in im around 0 66.8%

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

      1. add-cube-cbrt 65.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 65.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 65.5%

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

      6. neg-mul-1 65.5%

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

      7. *-commutative 65.5%

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

   7. Applied egg-rr 65.5%

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

      1. rem-cube-cbrt 66.8%

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

      2. distribute-rgt-neg-out 66.8%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.8%

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

   9. Applied egg-rr 66.8%

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

   if 3.39999999999999991 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 79.3%

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

      1. associate--r+ 79.3%

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

      2. unsub-neg 79.3%

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

      3. +-commutative 79.3%

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

      4. associate-+r- 79.3%

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

      5. neg-sub0 79.3%

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

      6. associate-+l- 79.3%

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

      7. sub0-neg 79.3%

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

      8. expm1-define 79.3%

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

   10. Simplified 79.3%

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

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.4:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\cos re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (cos re) -4e-310)
    im_m
    (*
     im_m
     (+
      (*
       im_m
       (-
        (* im_m (- (* im_m -0.020833333333333332) 0.08333333333333333))
        0.25))
      -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 (cos(re) <= -4e-310) {
		tmp = im_m;
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 (cos(re) <= (-4d-310)) then
        tmp = im_m
    else
        tmp = im_m * ((im_m * ((im_m * ((im_m * (-0.020833333333333332d0)) - 0.08333333333333333d0)) - 0.25d0)) + (-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 (Math.cos(re) <= -4e-310) {
		tmp = im_m;
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 math.cos(re) <= -4e-310:
		tmp = im_m
	else:
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 (cos(re) <= -4e-310)
		tmp = im_m;
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 (cos(re) <= -4e-310)
		tmp = im_m;
	else
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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[N[Cos[re], $MachinePrecision], -4e-310], im$95$m, N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -3.999999999999988e-310

   1. Initial program 53.8%

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

      1. /-rgt-identity 53.8%

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

      2. exp-0 53.8%

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

      3. associate-*l/ 53.8%

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

      4. cos-neg 53.8%

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

      5. associate-*l* 53.8%

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

      6. associate-*r/ 53.8%

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

      7. exp-0 53.8%

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

      8. /-rgt-identity 53.8%

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

      9. *-commutative 53.8%

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

      10. neg-sub0 53.8%

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

      11. cos-neg 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in re around 0 3.1%

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

   6. Taylor expanded in im around 0 2.5%

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

      1. *-rgt-identity 2.5%

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

      2. associate-*r* 2.5%

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

      3. metadata-eval 2.5%

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

      4. neg-mul-1 2.5%

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

      5. neg-sub0 2.5%

         \[\leadsto \color{blue}{0 - im} \]

      6. sub-neg 2.5%

         \[\leadsto \color{blue}{0 + \left(-im\right)} \]

      7. add-sqr-sqrt 0.9%

         \[\leadsto 0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}} \]

      8. sqrt-unprod 20.9%

         \[\leadsto 0 + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \]

      9. sqr-neg 20.9%

         \[\leadsto 0 + \sqrt{\color{blue}{im \cdot im}} \]

      10. sqrt-unprod 6.9%

          \[\leadsto 0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

      11. add-sqr-sqrt 13.0%

          \[\leadsto 0 + \color{blue}{im} \]

   8. Applied egg-rr 13.0%

      \[\leadsto \color{blue}{0 + im} \]
   9. Step-by-step derivation

      1. +-lft-identity 13.0%

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

   10. Simplified 13.0%

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

   if -3.999999999999988e-310 < (cos.f64 re)

   1. Initial program 56.4%

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

      1. /-rgt-identity 56.4%

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

      2. exp-0 56.4%

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

      3. associate-*l/ 56.4%

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

      4. cos-neg 56.4%

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

      5. associate-*l* 56.4%

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

      6. associate-*r/ 56.4%

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

      7. exp-0 56.4%

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

      8. /-rgt-identity 56.4%

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

      9. *-commutative 56.4%

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

      10. neg-sub0 56.4%

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

      11. cos-neg 56.4%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in im around 0 31.5%

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

      1. neg-mul-1 31.5%

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

      2. unsub-neg 31.5%

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

   7. Simplified 31.5%

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

   8. Taylor expanded in re around 0 31.5%

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

      1. associate--r+ 31.5%

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

      2. unsub-neg 31.5%

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

      3. +-commutative 31.5%

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

      4. associate-+r- 37.7%

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

      5. neg-sub0 37.7%

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

      6. associate-+l- 37.7%

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

      7. sub0-neg 37.7%

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

      8. expm1-define 60.1%

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

   10. Simplified 60.1%

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

   11. Taylor expanded in im around 0 53.5%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(-0.020833333333333332 \cdot im - 0.08333333333333333\right) - 0.25\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 82000000:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(im\_m + \mathsf{expm1}\left(im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 82000000.0)
    (* (cos re) (- im_m))
    (if (<= im_m 2.2e+77)
      (* 0.5 (+ im_m (expm1 im_m)))
      (*
       im_m
       (+
        (*
         im_m
         (-
          (* im_m (- (* im_m -0.020833333333333332) 0.08333333333333333))
          0.25))
        -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 <= 82000000.0) {
		tmp = cos(re) * -im_m;
	} else if (im_m <= 2.2e+77) {
		tmp = 0.5 * (im_m + expm1(im_m));
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -1.0);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 82000000.0) {
		tmp = Math.cos(re) * -im_m;
	} else if (im_m <= 2.2e+77) {
		tmp = 0.5 * (im_m + Math.expm1(im_m));
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 82000000.0:
		tmp = math.cos(re) * -im_m
	elif im_m <= 2.2e+77:
		tmp = 0.5 * (im_m + math.expm1(im_m))
	else:
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 82000000.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	elseif (im_m <= 2.2e+77)
		tmp = Float64(0.5 * Float64(im_m + expm1(im_m)));
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 82000000.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], If[LessEqual[im$95$m, 2.2e+77], N[(0.5 * N[(im$95$m + N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 8.2e7

   1. Initial program 39.8%

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

      1. /-rgt-identity 39.8%

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

      2. exp-0 39.8%

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

      3. associate-*l/ 39.8%

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

      4. cos-neg 39.8%

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

      5. associate-*l* 39.8%

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

      6. associate-*r/ 39.8%

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

      7. exp-0 39.8%

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

      8. /-rgt-identity 39.8%

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

      9. *-commutative 39.8%

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

      10. neg-sub0 39.8%

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

      11. cos-neg 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in im around 0 66.5%

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

      1. add-cube-cbrt 65.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 65.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 65.2%

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

      6. neg-mul-1 65.2%

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

      7. *-commutative 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. rem-cube-cbrt 66.5%

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

      2. distribute-rgt-neg-out 66.5%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.5%

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

   9. Applied egg-rr 66.5%

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

   if 8.2e7 < im < 2.2e77

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 83.3%

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

      1. associate--r+ 83.3%

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

      2. unsub-neg 83.3%

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

      3. +-commutative 83.3%

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

      4. associate-+r- 83.3%

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

      5. neg-sub0 83.3%

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

      6. associate-+l- 83.3%

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

      7. sub0-neg 83.3%

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

      8. expm1-define 83.3%

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

   10. Simplified 83.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right)} \]
   11. Step-by-step derivation

       1. sub-neg 83.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) + \left(-im\right)\right)} \]

       2. distribute-lft-in 83.3%

          \[\leadsto \color{blue}{0.5 \cdot \left(-\mathsf{expm1}\left(im\right)\right) + 0.5 \cdot \left(-im\right)} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{-\mathsf{expm1}\left(im\right)} \cdot \sqrt{-\mathsf{expm1}\left(im\right)}\right)} + 0.5 \cdot \left(-im\right) \]

       4. sqrt-unprod 16.7%

          \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\left(-\mathsf{expm1}\left(im\right)\right) \cdot \left(-\mathsf{expm1}\left(im\right)\right)}} + 0.5 \cdot \left(-im\right) \]

       5. sqr-neg 16.7%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(im\right) \cdot \mathsf{expm1}\left(im\right)}} + 0.5 \cdot \left(-im\right) \]

       6. sqrt-unprod 16.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\mathsf{expm1}\left(im\right)} \cdot \sqrt{\mathsf{expm1}\left(im\right)}\right)} + 0.5 \cdot \left(-im\right) \]

       7. add-sqr-sqrt 16.7%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(im\right)} + 0.5 \cdot \left(-im\right) \]

       8. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot \color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \]

       9. sqrt-unprod 16.7%

          \[\leadsto 0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \]

       10. sqr-neg 16.7%

           \[\leadsto 0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot \sqrt{\color{blue}{im \cdot im}} \]

       11. sqrt-unprod 16.7%

           \[\leadsto 0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot \color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \]

       12. add-sqr-sqrt 16.7%

           \[\leadsto 0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot \color{blue}{im} \]

   12. Applied egg-rr 16.7%

       \[\leadsto \color{blue}{0.5 \cdot \mathsf{expm1}\left(im\right) + 0.5 \cdot im} \]
   13. Step-by-step derivation

       1. distribute-lft-out 16.7%

          \[\leadsto \color{blue}{0.5 \cdot \left(\mathsf{expm1}\left(im\right) + im\right)} \]

       2. +-commutative 16.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(im + \mathsf{expm1}\left(im\right)\right)} \]

   14. Simplified 16.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(im + \mathsf{expm1}\left(im\right)\right)} \]

   if 2.2e77 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 75.0%

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

      1. associate--r+ 75.0%

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

      2. unsub-neg 75.0%

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

      3. +-commutative 75.0%

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

      4. associate-+r- 75.0%

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

      5. neg-sub0 75.0%

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

      6. associate-+l- 75.0%

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

      7. sub0-neg 75.0%

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

      8. expm1-define 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in im around 0 75.0%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(-0.020833333333333332 \cdot im - 0.08333333333333333\right) - 0.25\right) - 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 82000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(im + \mathsf{expm1}\left(im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.1:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\_m\right)\right) - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5.1) (* (cos re) (- im_m)) (* 0.5 (- (- (expm1 im_m)) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.1) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = 0.5 * (-expm1(im_m) - 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 tmp;
	if (im_m <= 5.1) {
		tmp = Math.cos(re) * -im_m;
	} else {
		tmp = 0.5 * (-Math.expm1(im_m) - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.1:
		tmp = math.cos(re) * -im_m
	else:
		tmp = 0.5 * (-math.expm1(im_m) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.1)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(0.5 * Float64(Float64(-expm1(im_m)) - im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.1], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(0.5 * N[((-N[(Exp[im$95$m] - 1), $MachinePrecision]) - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.0999999999999996

   1. Initial program 39.4%

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

      1. /-rgt-identity 39.4%

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

      2. exp-0 39.4%

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

      3. associate-*l/ 39.4%

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

      4. cos-neg 39.4%

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

      5. associate-*l* 39.4%

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

      6. associate-*r/ 39.4%

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

      7. exp-0 39.4%

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

      8. /-rgt-identity 39.4%

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

      9. *-commutative 39.4%

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

      10. neg-sub0 39.4%

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

      11. cos-neg 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in im around 0 66.8%

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

      1. add-cube-cbrt 65.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 65.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 65.5%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 65.5%

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

      6. neg-mul-1 65.5%

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

      7. *-commutative 65.5%

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

   7. Applied egg-rr 65.5%

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

      1. rem-cube-cbrt 66.8%

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

      2. distribute-rgt-neg-out 66.8%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.8%

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

   9. Applied egg-rr 66.8%

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

   if 5.0999999999999996 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 76.8%

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

      1. associate--r+ 76.8%

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

      2. unsub-neg 76.8%

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

      3. +-commutative 76.8%

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

      4. associate-+r- 76.8%

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

      5. neg-sub0 76.8%

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

      6. associate-+l- 76.8%

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

      7. sub0-neg 76.8%

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

      8. expm1-define 76.8%

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

   10. Simplified 76.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.1:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 30000000000000:\\ \;\;\;\;\cos re \cdot \left(-im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 30000000000000.0)
    (* (cos re) (- im_m))
    (*
     im_m
     (+
      (*
       im_m
       (-
        (* im_m (- (* im_m -0.020833333333333332) 0.08333333333333333))
        0.25))
      -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 <= 30000000000000.0) {
		tmp = cos(re) * -im_m;
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 30000000000000.0d0) then
        tmp = cos(re) * -im_m
    else
        tmp = im_m * ((im_m * ((im_m * ((im_m * (-0.020833333333333332d0)) - 0.08333333333333333d0)) - 0.25d0)) + (-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 <= 30000000000000.0) {
		tmp = Math.cos(re) * -im_m;
	} else {
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 30000000000000.0:
		tmp = math.cos(re) * -im_m
	else:
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 30000000000000.0)
		tmp = Float64(cos(re) * Float64(-im_m));
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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 <= 30000000000000.0)
		tmp = cos(re) * -im_m;
	else
		tmp = im_m * ((im_m * ((im_m * ((im_m * -0.020833333333333332) - 0.08333333333333333)) - 0.25)) + -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, 30000000000000.0], N[(N[Cos[re], $MachinePrecision] * (-im$95$m)), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3e13

   1. Initial program 40.1%

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

      1. /-rgt-identity 40.1%

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

      2. exp-0 40.1%

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

      3. associate-*l/ 40.1%

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

      4. cos-neg 40.1%

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

      5. associate-*l* 40.1%

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

      6. associate-*r/ 40.1%

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

      7. exp-0 40.1%

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

      8. /-rgt-identity 40.1%

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

      9. *-commutative 40.1%

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

      10. neg-sub0 40.1%

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

      11. cos-neg 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in im around 0 66.1%

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

      1. add-cube-cbrt 64.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

      2. pow3 64.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

      3. associate-*r* 64.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

      4. associate-*r* 64.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

      5. metadata-eval 64.9%

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

      6. neg-mul-1 64.9%

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

      7. *-commutative 64.9%

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

   7. Applied egg-rr 64.9%

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

      1. rem-cube-cbrt 66.1%

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

      2. distribute-rgt-neg-out 66.1%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      3. distribute-lft-neg-in 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 3e13 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in re around 0 77.6%

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

      1. associate--r+ 77.6%

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

      2. unsub-neg 77.6%

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

      3. +-commutative 77.6%

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

      4. associate-+r- 77.6%

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

      5. neg-sub0 77.6%

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

      6. associate-+l- 77.6%

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

      7. sub0-neg 77.6%

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

      8. expm1-define 77.6%

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

   10. Simplified 77.6%

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

   11. Taylor expanded in im around 0 63.4%

       \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(im \cdot \left(-0.020833333333333332 \cdot im - 0.08333333333333333\right) - 0.25\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 30000000000000:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.020833333333333332 - 0.08333333333333333\right) - 0.25\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.08333333333333333 - 0.25\right) + -1\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (+ (* im_m (- (* im_m -0.08333333333333333) 0.25)) -1.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * ((im_m * -0.08333333333333333) - 0.25)) + -1.0));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * ((im_m * ((im_m * (-0.08333333333333333d0)) - 0.25d0)) + (-1.0d0)))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * ((im_m * -0.08333333333333333) - 0.25)) + -1.0));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * ((im_m * ((im_m * -0.08333333333333333) - 0.25)) + -1.0))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.08333333333333333) - 0.25)) + -1.0)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * ((im_m * ((im_m * -0.08333333333333333) - 0.25)) + -1.0));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.08333333333333333), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in im around 0 31.6%

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

   1. neg-mul-1 31.6%

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

   2. unsub-neg 31.6%

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

7. Simplified 31.6%

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

8. Taylor expanded in re around 0 24.8%

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

   1. associate--r+ 24.8%

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

   2. unsub-neg 24.8%

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

   3. +-commutative 24.8%

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

   4. associate-+r- 29.3%

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

   5. neg-sub0 29.3%

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

   6. associate-+l- 29.3%

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

   7. sub0-neg 29.3%

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

   8. expm1-define 46.3%

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

10. Simplified 46.3%

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

11. Taylor expanded in im around 0 50.9%

    \[\leadsto \color{blue}{im \cdot \left(im \cdot \left(-0.08333333333333333 \cdot im - 0.25\right) - 1\right)} \]

12. Final simplification 50.9%

    \[\leadsto im \cdot \left(im \cdot \left(im \cdot -0.08333333333333333 - 0.25\right) + -1\right) \]
13. Add Preprocessing

Alternative 9: 46.9% accurate, 44.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(im\_m \cdot -0.25 + -1\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (+ (* im_m -0.25) -1.0))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * -0.25) + -1.0));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * ((im_m * (-0.25d0)) + (-1.0d0)))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * -0.25) + -1.0));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * ((im_m * -0.25) + -1.0))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(Float64(im_m * -0.25) + -1.0)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * ((im_m * -0.25) + -1.0));
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in im around 0 31.6%

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

   1. neg-mul-1 31.6%

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

   2. unsub-neg 31.6%

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

7. Simplified 31.6%

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

8. Taylor expanded in re around 0 24.8%

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

   1. associate--r+ 24.8%

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

   2. unsub-neg 24.8%

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

   3. +-commutative 24.8%

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

   4. associate-+r- 29.3%

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

   5. neg-sub0 29.3%

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

   6. associate-+l- 29.3%

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

   7. sub0-neg 29.3%

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

   8. expm1-define 46.3%

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

10. Simplified 46.3%

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

11. Taylor expanded in im around 0 39.1%

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

12. Final simplification 39.1%

    \[\leadsto im \cdot \left(im \cdot -0.25 + -1\right) \]
13. Add Preprocessing

Alternative 10: 29.6% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in im around 0 50.4%

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

   1. add-cube-cbrt 49.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)} \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right) \cdot \sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}} \]

   2. pow3 49.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)}\right)}^{3}} \]

   3. associate-*r* 49.4%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}}\right)}^{3} \]

   4. associate-*r* 49.4%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re}\right)}^{3} \]

   5. metadata-eval 49.4%

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

   6. neg-mul-1 49.4%

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

   7. *-commutative 49.4%

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

7. Applied egg-rr 49.4%

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

8. Taylor expanded in re around 0 27.1%

   \[\leadsto \color{blue}{im \cdot {\left(\sqrt[3]{-1}\right)}^{3}} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 27.1%

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

   2. *-commutative 27.1%

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

   3. neg-mul-1 27.1%

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

10. Simplified 27.1%

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

Alternative 11: 5.0% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 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 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in re around 0 43.7%

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

6. Taylor expanded in im around 0 27.1%

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

   1. *-rgt-identity 27.1%

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

   2. associate-*r* 27.1%

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

   3. metadata-eval 27.1%

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

   4. neg-mul-1 27.1%

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

   5. neg-sub0 27.1%

      \[\leadsto \color{blue}{0 - im} \]

   6. sub-neg 27.1%

      \[\leadsto \color{blue}{0 + \left(-im\right)} \]

   7. add-sqr-sqrt 14.6%

      \[\leadsto 0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}} \]

   8. sqrt-unprod 20.4%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \]

   9. sqr-neg 20.4%

      \[\leadsto 0 + \sqrt{\color{blue}{im \cdot im}} \]

   10. sqrt-unprod 2.3%

       \[\leadsto 0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}} \]

   11. add-sqr-sqrt 4.6%

       \[\leadsto 0 + \color{blue}{im} \]

8. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{0 + im} \]
9. Step-by-step derivation

   1. +-lft-identity 4.6%

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

10. Simplified 4.6%

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

Alternative 12: 3.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 0 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * 0.0d0
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

6. Applied egg-rr 3.5%

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

7. Taylor expanded in re around 0 3.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Alternative 13: 3.6% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. /-rgt-identity 55.8%

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

   2. exp-0 55.8%

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

   3. associate-*l/ 55.8%

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

   4. cos-neg 55.8%

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

   5. associate-*l* 55.8%

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

   6. associate-*r/ 55.8%

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

   7. exp-0 55.8%

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

   8. /-rgt-identity 55.8%

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

   9. *-commutative 55.8%

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

   10. neg-sub0 55.8%

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

   11. cos-neg 55.8%

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

3. Simplified 55.8%

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

6. Applied egg-rr 2.8%

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

7. Taylor expanded in re around 0 2.8%

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

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024111 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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