math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.0%

Time: 11.6s

Alternatives: 12

Speedup: 2.8×

• 50.3% 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: 54.7% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot \left(-2 \cdot \cos re\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* 0.5 (log1p (expm1 (* im_m (* -2.0 (cos re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * log1p(expm1((im_m * (-2.0 * cos(re))))));
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * Math.log1p(Math.expm1((im_m * (-2.0 * Math.cos(re))))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * math.log1p(math.expm1((im_m * (-2.0 * math.cos(re))))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * log1p(expm1(Float64(im_m * Float64(-2.0 * cos(re)))))))
end

Wolfram

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

Derivation

1. Initial program 58.3%

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

   1. /-rgt-identity 58.3%

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

   2. exp-0 58.3%

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

   3. associate-*l/ 58.3%

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

   4. cos-neg 58.3%

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

   5. associate-*l* 58.3%

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

   6. associate-*r/ 58.3%

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

   7. exp-0 58.3%

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

   8. /-rgt-identity 58.3%

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

   9. *-commutative 58.3%

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

   10. neg-sub0 58.3%

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

   11. cos-neg 58.3%

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

3. Simplified 58.3%

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

5. Taylor expanded in im around 0 47.8%

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

   1. log1p-expm1-u 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 440:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot {im\_m}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left({im\_m}^{7}\right)\right) \cdot -0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(-0.0003968253968253968 \cdot {im\_m}^{6} - 2\right)\right)\right)\\ \end{array} \end{array} \]

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 440.0)
    (*
     0.5
     (* (cos re) (* im_m (- (* -0.3333333333333333 (pow im_m 2.0)) 2.0))))
    (if (<= im_m 4.2e+40)
      (* 0.5 (* (log1p (expm1 (pow im_m 7.0))) -0.0003968253968253968))
      (*
       0.5
       (*
        (cos re)
        (* im_m (- (* -0.0003968253968253968 (pow im_m 6.0)) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 440.0) {
		tmp = 0.5 * (cos(re) * (im_m * ((-0.3333333333333333 * pow(im_m, 2.0)) - 2.0)));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (log1p(expm1(pow(im_m, 7.0))) * -0.0003968253968253968);
	} else {
		tmp = 0.5 * (cos(re) * (im_m * ((-0.0003968253968253968 * pow(im_m, 6.0)) - 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 <= 440.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((-0.3333333333333333 * Math.pow(im_m, 2.0)) - 2.0)));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (Math.log1p(Math.expm1(Math.pow(im_m, 7.0))) * -0.0003968253968253968);
	} else {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((-0.0003968253968253968 * Math.pow(im_m, 6.0)) - 2.0)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 440.0:
		tmp = 0.5 * (math.cos(re) * (im_m * ((-0.3333333333333333 * math.pow(im_m, 2.0)) - 2.0)))
	elif im_m <= 4.2e+40:
		tmp = 0.5 * (math.log1p(math.expm1(math.pow(im_m, 7.0))) * -0.0003968253968253968)
	else:
		tmp = 0.5 * (math.cos(re) * (im_m * ((-0.0003968253968253968 * math.pow(im_m, 6.0)) - 2.0)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 440.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(-0.3333333333333333 * (im_m ^ 2.0)) - 2.0))));
	elseif (im_m <= 4.2e+40)
		tmp = Float64(0.5 * Float64(log1p(expm1((im_m ^ 7.0))) * -0.0003968253968253968));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(-0.0003968253968253968 * (im_m ^ 6.0)) - 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, 440.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[Power[im$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.2e+40], N[(0.5 * N[(N[Log[1 + N[(Exp[N[Power[im$95$m, 7.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(-0.0003968253968253968 * N[Power[im$95$m, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 440

   1. Initial program 41.4%

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

      1. /-rgt-identity 41.4%

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

      2. exp-0 41.4%

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

      3. associate-*l/ 41.4%

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

      4. cos-neg 41.4%

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

      5. associate-*l* 41.4%

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

      6. associate-*r/ 41.4%

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

      7. exp-0 41.4%

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

      8. /-rgt-identity 41.4%

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

      9. *-commutative 41.4%

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

      10. neg-sub0 41.4%

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

      11. cos-neg 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in im around 0 88.0%

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

   if 440 < im < 4.2000000000000002e40

   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 5.3%

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

   6. Taylor expanded in im around inf 5.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 4.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 4.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 4.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 75.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{7}\right)\right)} \cdot -0.0003968253968253968\right) \]

   11. Applied egg-rr 75.0%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{7}\right)\right)} \cdot -0.0003968253968253968\right) \]

   if 4.2000000000000002e40 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.6%

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

   6. Taylor expanded in im around inf 98.6%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 440:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left({im}^{7}\right)\right) \cdot -0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im\_m \leq 7.5 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \sqrt{{im\_m}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.0003968253968253968 \cdot {im\_m}^{6} - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 460.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (<= im_m 4.1e+40)
      (* 0.5 (* im_m (* -0.08333333333333333 (pow re 4.0))))
      (if (<= im_m 7.5e+73)
        (* 0.5 (sqrt (* (pow im_m 14.0) 1.5747039556563367e-7)))
        (* 0.5 (* im_m (- (* -0.0003968253968253968 (pow im_m 6.0)) 2.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 460.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if (im_m <= 4.1e+40) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * pow(re, 4.0)));
	} else if (im_m <= 7.5e+73) {
		tmp = 0.5 * sqrt((pow(im_m, 14.0) * 1.5747039556563367e-7));
	} else {
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * pow(im_m, 6.0)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 460.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if (im_m <= 4.1d+40) then
        tmp = 0.5d0 * (im_m * ((-0.08333333333333333d0) * (re ** 4.0d0)))
    else if (im_m <= 7.5d+73) then
        tmp = 0.5d0 * sqrt(((im_m ** 14.0d0) * 1.5747039556563367d-7))
    else
        tmp = 0.5d0 * (im_m * (((-0.0003968253968253968d0) * (im_m ** 6.0d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if (im_m <= 4.1e+40) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * Math.pow(re, 4.0)));
	} else if (im_m <= 7.5e+73) {
		tmp = 0.5 * Math.sqrt((Math.pow(im_m, 14.0) * 1.5747039556563367e-7));
	} else {
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * Math.pow(im_m, 6.0)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 460.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif im_m <= 4.1e+40:
		tmp = 0.5 * (im_m * (-0.08333333333333333 * math.pow(re, 4.0)))
	elif im_m <= 7.5e+73:
		tmp = 0.5 * math.sqrt((math.pow(im_m, 14.0) * 1.5747039556563367e-7))
	else:
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * math.pow(im_m, 6.0)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif (im_m <= 4.1e+40)
		tmp = Float64(0.5 * Float64(im_m * Float64(-0.08333333333333333 * (re ^ 4.0))));
	elseif (im_m <= 7.5e+73)
		tmp = Float64(0.5 * sqrt(Float64((im_m ^ 14.0) * 1.5747039556563367e-7)));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 460.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif (im_m <= 4.1e+40)
		tmp = 0.5 * (im_m * (-0.08333333333333333 * (re ^ 4.0)));
	elseif (im_m <= 7.5e+73)
		tmp = 0.5 * sqrt(((im_m ^ 14.0) * 1.5747039556563367e-7));
	else
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.1e+40], N[(0.5 * N[(im$95$m * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 7.5e+73], N[(0.5 * N[Sqrt[N[(N[Power[im$95$m, 14.0], $MachinePrecision] * 1.5747039556563367e-7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(-0.0003968253968253968 * N[Power[im$95$m, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 460

   1. Initial program 41.4%

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

      1. /-rgt-identity 41.4%

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

      2. exp-0 41.4%

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

      3. associate-*l/ 41.4%

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

      4. cos-neg 41.4%

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

      5. associate-*l* 41.4%

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

      6. associate-*r/ 41.4%

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

      7. exp-0 41.4%

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

      8. /-rgt-identity 41.4%

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

      9. *-commutative 41.4%

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

      10. neg-sub0 41.4%

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

      11. cos-neg 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in im around 0 65.1%

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

   if 460 < im < 4.1000000000000002e40

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

   6. Taylor expanded in re around 0 50.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot -2} + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      2. distribute-rgt-in 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot -2 + \color{blue}{\left(im \cdot {re}^{2} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)}\right) \]

      3. associate-+r+ 25.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot -2 + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)} \]

      4. distribute-lft-out 25.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      5. associate-*r* 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{2}\right)} \cdot {re}^{2}\right) \]

      6. associate-*l* 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot \left({re}^{2} \cdot {re}^{2}\right)}\right) \]

      7. *-commutative 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) \]

      8. pow-sqr 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot \color{blue}{{re}^{\left(2 \cdot 2\right)}}\right) \]

      9. metadata-eval 25.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{\color{blue}{4}}\right) \]

   8. Simplified 25.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{4}\right)} \]

   9. Taylor expanded in re around inf 50.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 50.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}\right)} \]

       2. *-commutative 50.5%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

       3. associate-*l* 50.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   11. Simplified 50.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   if 4.1000000000000002e40 < im < 7.5e73

   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 88.6%

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

   6. Taylor expanded in im around inf 88.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 25.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 25.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 25.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{im}^{7} \cdot -0.0003968253968253968} \cdot \sqrt{{im}^{7} \cdot -0.0003968253968253968}\right)} \]

       2. sqrt-unprod 75.0%

          \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\left({im}^{7} \cdot -0.0003968253968253968\right) \cdot \left({im}^{7} \cdot -0.0003968253968253968\right)}} \]

       3. swap-sqr 75.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left({im}^{7} \cdot {im}^{7}\right) \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)}} \]

       4. pow-prod-up 75.0%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{im}^{\left(7 + 7\right)}} \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)} \]

       5. metadata-eval 75.0%

          \[\leadsto 0.5 \cdot \sqrt{{im}^{\color{blue}{14}} \cdot \left(-0.0003968253968253968 \cdot -0.0003968253968253968\right)} \]

       6. metadata-eval 75.0%

          \[\leadsto 0.5 \cdot \sqrt{{im}^{14} \cdot \color{blue}{1.5747039556563367 \cdot 10^{-7}}} \]

   11. Applied egg-rr 75.0%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{im}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}}} \]

   if 7.5e73 < 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%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 70.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \sqrt{{im}^{14} \cdot 1.5747039556563367 \cdot 10^{-7}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 68000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im\_m}^{14}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im\_m}^{7}\right)\right)\\ \end{array} \end{array} \]

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 68000000000.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (<= im_m 4.2e+40)
      (* 0.5 (* -0.0003968253968253968 (sqrt (pow im_m 14.0))))
      (* 0.5 (* -0.0003968253968253968 (* (cos re) (pow im_m 7.0))))))))

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 <= 68000000000.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (-0.0003968253968253968 * sqrt(pow(im_m, 14.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * pow(im_m, 7.0)));
	}
	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 <= 68000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if (im_m <= 4.2d+40) then
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * sqrt((im_m ** 14.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (cos(re) * (im_m ** 7.0d0)))
    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 <= 68000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (-0.0003968253968253968 * Math.sqrt(Math.pow(im_m, 14.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(re) * Math.pow(im_m, 7.0)));
	}
	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 <= 68000000000.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif im_m <= 4.2e+40:
		tmp = 0.5 * (-0.0003968253968253968 * math.sqrt(math.pow(im_m, 14.0)))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(re) * math.pow(im_m, 7.0)))
	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 <= 68000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif (im_m <= 4.2e+40)
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * sqrt((im_m ^ 14.0))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(re) * (im_m ^ 7.0))));
	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 <= 68000000000.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif (im_m <= 4.2e+40)
		tmp = 0.5 * (-0.0003968253968253968 * sqrt((im_m ^ 14.0)));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * (im_m ^ 7.0)));
	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, 68000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.2e+40], N[(0.5 * N[(-0.0003968253968253968 * N[Sqrt[N[Power[im$95$m, 14.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.8e10

   1. Initial program 42.0%

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

      1. /-rgt-identity 42.0%

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

      2. exp-0 42.0%

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

      3. associate-*l/ 42.0%

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

      4. cos-neg 42.0%

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

      5. associate-*l* 42.0%

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

      6. associate-*r/ 42.0%

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

      7. exp-0 42.0%

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

      8. /-rgt-identity 42.0%

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

      9. *-commutative 42.0%

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

      10. neg-sub0 42.0%

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

      11. cos-neg 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in im around 0 64.4%

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

   if 6.8e10 < im < 4.2000000000000002e40

   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 5.9%

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

   6. Taylor expanded in im around inf 5.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 4.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 4.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{im}^{7}} \cdot \sqrt{{im}^{7}}\right)} \cdot -0.0003968253968253968\right) \]

       2. sqrt-unprod 51.4%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{7} \cdot {im}^{7}}} \cdot -0.0003968253968253968\right) \]

       3. pow-prod-up 51.4%

          \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{im}^{\left(7 + 7\right)}}} \cdot -0.0003968253968253968\right) \]

       4. metadata-eval 51.4%

          \[\leadsto 0.5 \cdot \left(\sqrt{{im}^{\color{blue}{14}}} \cdot -0.0003968253968253968\right) \]

   11. Applied egg-rr 51.4%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{14}}} \cdot -0.0003968253968253968\right) \]

   if 4.2000000000000002e40 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.6%

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

   6. Taylor expanded in im around inf 98.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 68000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im}^{14}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 70000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot {im\_m}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im\_m}^{14}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im\_m}^{7}\right)\right)\\ \end{array} \end{array} \]

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 70000000000.0)
    (*
     0.5
     (* (cos re) (* im_m (- (* -0.3333333333333333 (pow im_m 2.0)) 2.0))))
    (if (<= im_m 4.2e+40)
      (* 0.5 (* -0.0003968253968253968 (sqrt (pow im_m 14.0))))
      (* 0.5 (* -0.0003968253968253968 (* (cos re) (pow im_m 7.0))))))))

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 <= 70000000000.0) {
		tmp = 0.5 * (cos(re) * (im_m * ((-0.3333333333333333 * pow(im_m, 2.0)) - 2.0)));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (-0.0003968253968253968 * sqrt(pow(im_m, 14.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * pow(im_m, 7.0)));
	}
	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 <= 70000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (((-0.3333333333333333d0) * (im_m ** 2.0d0)) - 2.0d0)))
    else if (im_m <= 4.2d+40) then
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * sqrt((im_m ** 14.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * (cos(re) * (im_m ** 7.0d0)))
    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 <= 70000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * ((-0.3333333333333333 * Math.pow(im_m, 2.0)) - 2.0)));
	} else if (im_m <= 4.2e+40) {
		tmp = 0.5 * (-0.0003968253968253968 * Math.sqrt(Math.pow(im_m, 14.0)));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * (Math.cos(re) * Math.pow(im_m, 7.0)));
	}
	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 <= 70000000000.0:
		tmp = 0.5 * (math.cos(re) * (im_m * ((-0.3333333333333333 * math.pow(im_m, 2.0)) - 2.0)))
	elif im_m <= 4.2e+40:
		tmp = 0.5 * (-0.0003968253968253968 * math.sqrt(math.pow(im_m, 14.0)))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * (math.cos(re) * math.pow(im_m, 7.0)))
	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 <= 70000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(-0.3333333333333333 * (im_m ^ 2.0)) - 2.0))));
	elseif (im_m <= 4.2e+40)
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * sqrt((im_m ^ 14.0))));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * Float64(cos(re) * (im_m ^ 7.0))));
	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 <= 70000000000.0)
		tmp = 0.5 * (cos(re) * (im_m * ((-0.3333333333333333 * (im_m ^ 2.0)) - 2.0)));
	elseif (im_m <= 4.2e+40)
		tmp = 0.5 * (-0.0003968253968253968 * sqrt((im_m ^ 14.0)));
	else
		tmp = 0.5 * (-0.0003968253968253968 * (cos(re) * (im_m ^ 7.0)));
	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, 70000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[Power[im$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.2e+40], N[(0.5 * N[(-0.0003968253968253968 * N[Sqrt[N[Power[im$95$m, 14.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[(N[Cos[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7e10

   1. Initial program 42.0%

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

      1. /-rgt-identity 42.0%

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

      2. exp-0 42.0%

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

      3. associate-*l/ 42.0%

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

      4. cos-neg 42.0%

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

      5. associate-*l* 42.0%

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

      6. associate-*r/ 42.0%

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

      7. exp-0 42.0%

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

      8. /-rgt-identity 42.0%

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

      9. *-commutative 42.0%

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

      10. neg-sub0 42.0%

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

      11. cos-neg 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in im around 0 87.0%

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

   if 7e10 < im < 4.2000000000000002e40

   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 5.9%

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

   6. Taylor expanded in im around inf 5.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 4.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 4.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 4.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{im}^{7}} \cdot \sqrt{{im}^{7}}\right)} \cdot -0.0003968253968253968\right) \]

       2. sqrt-unprod 51.4%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{7} \cdot {im}^{7}}} \cdot -0.0003968253968253968\right) \]

       3. pow-prod-up 51.4%

          \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{im}^{\left(7 + 7\right)}}} \cdot -0.0003968253968253968\right) \]

       4. metadata-eval 51.4%

          \[\leadsto 0.5 \cdot \left(\sqrt{{im}^{\color{blue}{14}}} \cdot -0.0003968253968253968\right) \]

   11. Applied egg-rr 51.4%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{14}}} \cdot -0.0003968253968253968\right) \]

   if 4.2000000000000002e40 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.6%

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

   6. Taylor expanded in im around inf 98.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 70000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+40}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im}^{14}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot \left(-0.0003968253968253968 \cdot {im\_m}^{6} - 2\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   0.5
   (* (cos re) (* im_m (- (* -0.0003968253968253968 (pow im_m 6.0)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (cos(re) * (im_m * ((-0.0003968253968253968 * pow(im_m, 6.0)) - 2.0))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (0.5d0 * (cos(re) * (im_m * (((-0.0003968253968253968d0) * (im_m ** 6.0d0)) - 2.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (Math.cos(re) * (im_m * ((-0.0003968253968253968 * Math.pow(im_m, 6.0)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (math.cos(re) * (im_m * ((-0.0003968253968253968 * math.pow(im_m, 6.0)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(cos(re) * Float64(im_m * Float64(Float64(-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (cos(re) * (im_m * ((-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(N[(-0.0003968253968253968 * N[Power[im$95$m, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

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

   1. /-rgt-identity 58.3%

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

   2. exp-0 58.3%

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

   3. associate-*l/ 58.3%

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

   4. cos-neg 58.3%

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

   5. associate-*l* 58.3%

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

   6. associate-*r/ 58.3%

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

   7. exp-0 58.3%

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

   8. /-rgt-identity 58.3%

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

   9. *-commutative 58.3%

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

   10. neg-sub0 58.3%

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

   11. cos-neg 58.3%

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

3. Simplified 58.3%

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

5. Taylor expanded in im around 0 93.6%

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

6. Taylor expanded in im around inf 93.0%

   \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

7. Final simplification 93.0%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\right) \]
8. Add Preprocessing

Alternative 7: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 52000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im\_m}^{14}}\right)\\ \end{array} \end{array} \]

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 52000000000.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (* 0.5 (* -0.0003968253968253968 (sqrt (pow im_m 14.0)))))))

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 <= 52000000000.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * sqrt(pow(im_m, 14.0)));
	}
	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 <= 52000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else
        tmp = 0.5d0 * ((-0.0003968253968253968d0) * sqrt((im_m ** 14.0d0)))
    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 <= 52000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (-0.0003968253968253968 * Math.sqrt(Math.pow(im_m, 14.0)));
	}
	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 <= 52000000000.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	else:
		tmp = 0.5 * (-0.0003968253968253968 * math.sqrt(math.pow(im_m, 14.0)))
	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 <= 52000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	else
		tmp = Float64(0.5 * Float64(-0.0003968253968253968 * sqrt((im_m ^ 14.0))));
	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 <= 52000000000.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	else
		tmp = 0.5 * (-0.0003968253968253968 * sqrt((im_m ^ 14.0)));
	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, 52000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.0003968253968253968 * N[Sqrt[N[Power[im$95$m, 14.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.2e10

   1. Initial program 42.0%

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

      1. /-rgt-identity 42.0%

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

      2. exp-0 42.0%

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

      3. associate-*l/ 42.0%

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

      4. cos-neg 42.0%

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

      5. associate-*l* 42.0%

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

      6. associate-*r/ 42.0%

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

      7. exp-0 42.0%

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

      8. /-rgt-identity 42.0%

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

      9. *-commutative 42.0%

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

      10. neg-sub0 42.0%

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

      11. cos-neg 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in im around 0 64.4%

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

   if 5.2e10 < 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 90.9%

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

   6. Taylor expanded in im around inf 90.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 60.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 60.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 60.1%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{{im}^{7}} \cdot \sqrt{{im}^{7}}\right)} \cdot -0.0003968253968253968\right) \]

       2. sqrt-unprod 64.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{7} \cdot {im}^{7}}} \cdot -0.0003968253968253968\right) \]

       3. pow-prod-up 64.0%

          \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{im}^{\left(7 + 7\right)}}} \cdot -0.0003968253968253968\right) \]

       4. metadata-eval 64.0%

          \[\leadsto 0.5 \cdot \left(\sqrt{{im}^{\color{blue}{14}}} \cdot -0.0003968253968253968\right) \]

   11. Applied egg-rr 64.0%

       \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{{im}^{14}}} \cdot -0.0003968253968253968\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 52000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.0003968253968253968 \cdot \sqrt{{im}^{14}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.0003968253968253968 \cdot {im\_m}^{6} - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 460.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (<= im_m 1.5e+46)
      (* 0.5 (* im_m (* -0.08333333333333333 (pow re 4.0))))
      (* 0.5 (* im_m (- (* -0.0003968253968253968 (pow im_m 6.0)) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 460.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if (im_m <= 1.5e+46) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * pow(re, 4.0)));
	} else {
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * pow(im_m, 6.0)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 460.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if (im_m <= 1.5d+46) then
        tmp = 0.5d0 * (im_m * ((-0.08333333333333333d0) * (re ** 4.0d0)))
    else
        tmp = 0.5d0 * (im_m * (((-0.0003968253968253968d0) * (im_m ** 6.0d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if (im_m <= 1.5e+46) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * Math.pow(re, 4.0)));
	} else {
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * Math.pow(im_m, 6.0)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 460.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif im_m <= 1.5e+46:
		tmp = 0.5 * (im_m * (-0.08333333333333333 * math.pow(re, 4.0)))
	else:
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * math.pow(im_m, 6.0)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif (im_m <= 1.5e+46)
		tmp = Float64(0.5 * Float64(im_m * Float64(-0.08333333333333333 * (re ^ 4.0))));
	else
		tmp = Float64(0.5 * Float64(im_m * Float64(Float64(-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 460.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif (im_m <= 1.5e+46)
		tmp = 0.5 * (im_m * (-0.08333333333333333 * (re ^ 4.0)));
	else
		tmp = 0.5 * (im_m * ((-0.0003968253968253968 * (im_m ^ 6.0)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.5e+46], N[(0.5 * N[(im$95$m * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im$95$m * N[(N[(-0.0003968253968253968 * N[Power[im$95$m, 6.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 41.4%

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

      1. /-rgt-identity 41.4%

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

      2. exp-0 41.4%

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

      3. associate-*l/ 41.4%

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

      4. cos-neg 41.4%

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

      5. associate-*l* 41.4%

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

      6. associate-*r/ 41.4%

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

      7. exp-0 41.4%

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

      8. /-rgt-identity 41.4%

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

      9. *-commutative 41.4%

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

      10. neg-sub0 41.4%

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

      11. cos-neg 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in im around 0 65.1%

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

   if 460 < im < 1.50000000000000012e46

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.4%

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

   6. Taylor expanded in re around 0 37.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 37.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot -2} + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      2. distribute-rgt-in 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot -2 + \color{blue}{\left(im \cdot {re}^{2} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)}\right) \]

      3. associate-+r+ 18.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot -2 + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)} \]

      4. distribute-lft-out 18.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      5. associate-*r* 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{2}\right)} \cdot {re}^{2}\right) \]

      6. associate-*l* 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot \left({re}^{2} \cdot {re}^{2}\right)}\right) \]

      7. *-commutative 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) \]

      8. pow-sqr 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot \color{blue}{{re}^{\left(2 \cdot 2\right)}}\right) \]

      9. metadata-eval 18.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{\color{blue}{4}}\right) \]

   8. Simplified 18.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{4}\right)} \]

   9. Taylor expanded in re around inf 36.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 36.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}\right)} \]

       2. *-commutative 36.8%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

       3. associate-*l* 36.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   11. Simplified 36.8%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   if 1.50000000000000012e46 < 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%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(\color{blue}{-0.0003968253968253968 \cdot {im}^{6}} - 2\right)\right) \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 68.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.0003968253968253968 \cdot {im}^{6} - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{elif}\;im\_m \leq 9.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im\_m}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \end{array} \]

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 460.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (if (<= im_m 9.5e+46)
      (* 0.5 (* im_m (* -0.08333333333333333 (pow re 4.0))))
      (* 0.5 (* (pow im_m 7.0) -0.0003968253968253968))))))

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 <= 460.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else if (im_m <= 9.5e+46) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * pow(re, 4.0)));
	} else {
		tmp = 0.5 * (pow(im_m, 7.0) * -0.0003968253968253968);
	}
	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 <= 460.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else if (im_m <= 9.5d+46) then
        tmp = 0.5d0 * (im_m * ((-0.08333333333333333d0) * (re ** 4.0d0)))
    else
        tmp = 0.5d0 * ((im_m ** 7.0d0) * (-0.0003968253968253968d0))
    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 <= 460.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else if (im_m <= 9.5e+46) {
		tmp = 0.5 * (im_m * (-0.08333333333333333 * Math.pow(re, 4.0)));
	} else {
		tmp = 0.5 * (Math.pow(im_m, 7.0) * -0.0003968253968253968);
	}
	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 <= 460.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	elif im_m <= 9.5e+46:
		tmp = 0.5 * (im_m * (-0.08333333333333333 * math.pow(re, 4.0)))
	else:
		tmp = 0.5 * (math.pow(im_m, 7.0) * -0.0003968253968253968)
	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 <= 460.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	elseif (im_m <= 9.5e+46)
		tmp = Float64(0.5 * Float64(im_m * Float64(-0.08333333333333333 * (re ^ 4.0))));
	else
		tmp = Float64(0.5 * Float64((im_m ^ 7.0) * -0.0003968253968253968));
	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 <= 460.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	elseif (im_m <= 9.5e+46)
		tmp = 0.5 * (im_m * (-0.08333333333333333 * (re ^ 4.0)));
	else
		tmp = 0.5 * ((im_m ^ 7.0) * -0.0003968253968253968);
	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, 460.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.5e+46], N[(0.5 * N[(im$95$m * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 41.4%

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

      1. /-rgt-identity 41.4%

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

      2. exp-0 41.4%

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

      3. associate-*l/ 41.4%

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

      4. cos-neg 41.4%

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

      5. associate-*l* 41.4%

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

      6. associate-*r/ 41.4%

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

      7. exp-0 41.4%

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

      8. /-rgt-identity 41.4%

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

      9. *-commutative 41.4%

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

      10. neg-sub0 41.4%

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

      11. cos-neg 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in im around 0 65.1%

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

   if 460 < im < 9.5000000000000008e46

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.4%

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

   6. Taylor expanded in re around 0 34.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 34.0%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot -2} + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      2. distribute-rgt-in 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot -2 + \color{blue}{\left(im \cdot {re}^{2} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)}\right) \]

      3. associate-+r+ 17.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot -2 + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)} \]

      4. distribute-lft-out 17.3%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      5. associate-*r* 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{2}\right)} \cdot {re}^{2}\right) \]

      6. associate-*l* 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot \left({re}^{2} \cdot {re}^{2}\right)}\right) \]

      7. *-commutative 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) \]

      8. pow-sqr 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot \color{blue}{{re}^{\left(2 \cdot 2\right)}}\right) \]

      9. metadata-eval 17.3%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{\color{blue}{4}}\right) \]

   8. Simplified 17.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{4}\right)} \]

   9. Taylor expanded in re around inf 33.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 33.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}\right)} \]

       2. *-commutative 33.7%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

       3. associate-*l* 33.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   11. Simplified 33.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)} \]

   if 9.5000000000000008e46 < 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%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 69.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 69.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 66000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im\_m \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im\_m}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \end{array} \]

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 66000000000.0)
    (* 0.5 (* (cos re) (* im_m -2.0)))
    (* 0.5 (* (pow im_m 7.0) -0.0003968253968253968)))))

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 <= 66000000000.0) {
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (pow(im_m, 7.0) * -0.0003968253968253968);
	}
	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 <= 66000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im_m * (-2.0d0)))
    else
        tmp = 0.5d0 * ((im_m ** 7.0d0) * (-0.0003968253968253968d0))
    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 <= 66000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im_m * -2.0));
	} else {
		tmp = 0.5 * (Math.pow(im_m, 7.0) * -0.0003968253968253968);
	}
	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 <= 66000000000.0:
		tmp = 0.5 * (math.cos(re) * (im_m * -2.0))
	else:
		tmp = 0.5 * (math.pow(im_m, 7.0) * -0.0003968253968253968)
	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 <= 66000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im_m * -2.0)));
	else
		tmp = Float64(0.5 * Float64((im_m ^ 7.0) * -0.0003968253968253968));
	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 <= 66000000000.0)
		tmp = 0.5 * (cos(re) * (im_m * -2.0));
	else
		tmp = 0.5 * ((im_m ^ 7.0) * -0.0003968253968253968);
	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, 66000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.6e10

   1. Initial program 42.0%

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

      1. /-rgt-identity 42.0%

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

      2. exp-0 42.0%

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

      3. associate-*l/ 42.0%

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

      4. cos-neg 42.0%

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

      5. associate-*l* 42.0%

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

      6. associate-*r/ 42.0%

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

      7. exp-0 42.0%

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

      8. /-rgt-identity 42.0%

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

      9. *-commutative 42.0%

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

      10. neg-sub0 42.0%

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

      11. cos-neg 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in im around 0 64.4%

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

   if 6.6e10 < 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 90.9%

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

   6. Taylor expanded in im around inf 90.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 60.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 60.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 66000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.1:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im\_m}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.1)
    (* 0.5 (* im_m -2.0))
    (* 0.5 (* (pow im_m 7.0) -0.0003968253968253968)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.1) {
		tmp = 0.5 * (im_m * -2.0);
	} else {
		tmp = 0.5 * (pow(im_m, 7.0) * -0.0003968253968253968);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.1d0) then
        tmp = 0.5d0 * (im_m * (-2.0d0))
    else
        tmp = 0.5d0 * ((im_m ** 7.0d0) * (-0.0003968253968253968d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.1) {
		tmp = 0.5 * (im_m * -2.0);
	} else {
		tmp = 0.5 * (Math.pow(im_m, 7.0) * -0.0003968253968253968);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.1:
		tmp = 0.5 * (im_m * -2.0)
	else:
		tmp = 0.5 * (math.pow(im_m, 7.0) * -0.0003968253968253968)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.1)
		tmp = Float64(0.5 * Float64(im_m * -2.0));
	else
		tmp = Float64(0.5 * Float64((im_m ^ 7.0) * -0.0003968253968253968));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.1)
		tmp = 0.5 * (im_m * -2.0);
	else
		tmp = 0.5 * ((im_m ^ 7.0) * -0.0003968253968253968);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.1], N[(0.5 * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[im$95$m, 7.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.10000000000000001

   1. Initial program 41.4%

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

      1. /-rgt-identity 41.4%

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

      2. exp-0 41.4%

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

      3. associate-*l/ 41.4%

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

      4. cos-neg 41.4%

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

      5. associate-*l* 41.4%

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

      6. associate-*r/ 41.4%

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

      7. exp-0 41.4%

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

      8. /-rgt-identity 41.4%

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

      9. *-commutative 41.4%

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

      10. neg-sub0 41.4%

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

      11. cos-neg 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in im around 0 65.1%

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

   6. Taylor expanded in re around 0 35.6%

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

      1. *-commutative 35.6%

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

   8. Simplified 35.6%

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

   if 0.10000000000000001 < 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 88.5%

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

   6. Taylor expanded in im around inf 88.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]

   7. Taylor expanded in re around 0 58.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 58.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{7} \cdot -0.0003968253968253968\right)} \]

   9. Simplified 58.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.1:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({im}^{7} \cdot -0.0003968253968253968\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.0% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

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

   1. /-rgt-identity 58.3%

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

   2. exp-0 58.3%

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

   3. associate-*l/ 58.3%

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

   4. cos-neg 58.3%

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

   5. associate-*l* 58.3%

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

   6. associate-*r/ 58.3%

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

   7. exp-0 58.3%

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

   8. /-rgt-identity 58.3%

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

   9. *-commutative 58.3%

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

   10. neg-sub0 58.3%

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

   11. cos-neg 58.3%

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

3. Simplified 58.3%

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

5. Taylor expanded in im around 0 47.8%

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

6. Taylor expanded in re around 0 26.4%

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

   1. *-commutative 26.4%

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

8. Simplified 26.4%

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

9. Final simplification 26.4%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]
10. 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 2024069 
(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))))