math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.0%

Time: 10.2s

Alternatives: 8

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% 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

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

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* 0.5 (log1p (expm1 (* -2.0 (* im_m (cos re))))))))

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. cos-neg 50.3%

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

   2. sub-neg 50.3%

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

   3. neg-sub0 50.3%

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

   4. remove-double-neg 50.3%

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

   5. remove-double-neg 50.3%

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

   6. sub0-neg 50.3%

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

   7. distribute-neg-in 50.3%

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

   8. +-commutative 50.3%

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

   9. sub-neg 50.3%

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

   10. associate-*l* 50.3%

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

   11. sub-neg 50.3%

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

   12. +-commutative 50.3%

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

   13. distribute-neg-in 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in im around 0 56.7%

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

   1. log1p-expm1-u 99.0%

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

   2. associate-*l* 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 2: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 400:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im_m + -0.3333333333333333 \cdot {im_m}^{3}\right)\right)\\ \mathbf{elif}\;im_m \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 400.0)
    (*
     0.5
     (* (cos re) (+ (* -2.0 im_m) (* -0.3333333333333333 (pow im_m 3.0)))))
    (if (<= im_m 2.45e+95)
      (* 0.5 (log1p (expm1 (* -2.0 im_m))))
      (* (cos re) (* (pow im_m 3.0) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 400.0) {
		tmp = 0.5 * (cos(re) * ((-2.0 * im_m) + (-0.3333333333333333 * pow(im_m, 3.0))));
	} else if (im_m <= 2.45e+95) {
		tmp = 0.5 * log1p(expm1((-2.0 * im_m)));
	} else {
		tmp = cos(re) * (pow(im_m, 3.0) * -0.16666666666666666);
	}
	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 <= 400.0) {
		tmp = 0.5 * (Math.cos(re) * ((-2.0 * im_m) + (-0.3333333333333333 * Math.pow(im_m, 3.0))));
	} else if (im_m <= 2.45e+95) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im_m)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 400.0:
		tmp = 0.5 * (math.cos(re) * ((-2.0 * im_m) + (-0.3333333333333333 * math.pow(im_m, 3.0))))
	elif im_m <= 2.45e+95:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im_m)))
	else:
		tmp = math.cos(re) * (math.pow(im_m, 3.0) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 400.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(-2.0 * im_m) + Float64(-0.3333333333333333 * (im_m ^ 3.0)))));
	elseif (im_m <= 2.45e+95)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im_m))));
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 3.0) * -0.16666666666666666));
	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, 400.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(-2.0 * im$95$m), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+95], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im$95$m), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 400

   1. Initial program 33.7%

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

      1. cos-neg 33.7%

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

      2. sub-neg 33.7%

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

      3. neg-sub0 33.7%

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

      4. remove-double-neg 33.7%

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

      5. remove-double-neg 33.7%

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

      6. sub0-neg 33.7%

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

      7. distribute-neg-in 33.7%

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

      8. +-commutative 33.7%

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

      9. sub-neg 33.7%

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

      10. associate-*l* 33.7%

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

      11. sub-neg 33.7%

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

      12. +-commutative 33.7%

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

      13. distribute-neg-in 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in im around 0 93.6%

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

   if 400 < im < 2.4499999999999999e95

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 3.6%

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

      1. log1p-expm1-u 93.7%

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

      2. associate-*l* 93.7%

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

   7. Applied egg-rr 93.7%

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

   8. Taylor expanded in re around 0 93.7%

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

   if 2.4499999999999999e95 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.1%

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

   6. Taylor expanded in im around inf 98.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.165:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im_m\right)\right)\\ \mathbf{elif}\;im_m \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im_m}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.165)
    (* 0.5 (* (cos re) (* -2.0 im_m)))
    (if (<= im_m 2.45e+95)
      (* 0.5 (log1p (expm1 (* -2.0 im_m))))
      (* (cos re) (* (pow im_m 3.0) -0.16666666666666666))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.165) {
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	} else if (im_m <= 2.45e+95) {
		tmp = 0.5 * log1p(expm1((-2.0 * im_m)));
	} else {
		tmp = cos(re) * (pow(im_m, 3.0) * -0.16666666666666666);
	}
	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 <= 0.165) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im_m));
	} else if (im_m <= 2.45e+95) {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im_m)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im_m, 3.0) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.165:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im_m))
	elif im_m <= 2.45e+95:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im_m)))
	else:
		tmp = math.cos(re) * (math.pow(im_m, 3.0) * -0.16666666666666666)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.165)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im_m)));
	elseif (im_m <= 2.45e+95)
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im_m))));
	else
		tmp = Float64(cos(re) * Float64((im_m ^ 3.0) * -0.16666666666666666));
	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, 0.165], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+95], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im$95$m), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.165000000000000008

   1. Initial program 33.7%

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

      1. cos-neg 33.7%

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

      2. sub-neg 33.7%

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

      3. neg-sub0 33.7%

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

      4. remove-double-neg 33.7%

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

      5. remove-double-neg 33.7%

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

      6. sub0-neg 33.7%

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

      7. distribute-neg-in 33.7%

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

      8. +-commutative 33.7%

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

      9. sub-neg 33.7%

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

      10. associate-*l* 33.7%

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

      11. sub-neg 33.7%

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

      12. +-commutative 33.7%

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

      13. distribute-neg-in 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in im around 0 73.6%

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

   if 0.165000000000000008 < im < 2.4499999999999999e95

   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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 3.6%

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

      1. log1p-expm1-u 93.7%

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

      2. associate-*l* 93.7%

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

   7. Applied egg-rr 93.7%

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

   8. Taylor expanded in re around 0 93.7%

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

   if 2.4499999999999999e95 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 98.1%

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

   6. Taylor expanded in im around inf 98.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in im around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.165:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.165:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.165)
    (* 0.5 (* (cos re) (* -2.0 im_m)))
    (* 0.5 (log1p (expm1 (* -2.0 im_m)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.165) {
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	} else {
		tmp = 0.5 * log1p(expm1((-2.0 * im_m)));
	}
	return im_s * tmp;
}

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.165) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im_m));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((-2.0 * im_m)));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.165:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im_m))
	else:
		tmp = 0.5 * math.log1p(math.expm1((-2.0 * im_m)))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.165)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im_m)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(-2.0 * im_m))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.165], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(-2.0 * im$95$m), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.165000000000000008

   1. Initial program 33.7%

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

      1. cos-neg 33.7%

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

      2. sub-neg 33.7%

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

      3. neg-sub0 33.7%

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

      4. remove-double-neg 33.7%

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

      5. remove-double-neg 33.7%

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

      6. sub0-neg 33.7%

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

      7. distribute-neg-in 33.7%

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

      8. +-commutative 33.7%

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

      9. sub-neg 33.7%

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

      10. associate-*l* 33.7%

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

      11. sub-neg 33.7%

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

      12. +-commutative 33.7%

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

      13. distribute-neg-in 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in im around 0 73.6%

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

   if 0.165000000000000008 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 6.0%

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

      1. log1p-expm1-u 98.5%

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

      2. associate-*l* 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in re around 0 78.2%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot \color{blue}{im}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.165:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-2 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.00072:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im_m + -0.3333333333333333 \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00072)
    (* 0.5 (* (cos re) (* -2.0 im_m)))
    (* 0.5 (+ (* -2.0 im_m) (* -0.3333333333333333 (pow im_m 3.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00072) {
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	} else {
		tmp = 0.5 * ((-2.0 * im_m) + (-0.3333333333333333 * pow(im_m, 3.0)));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.00072d0) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im_m))
    else
        tmp = 0.5d0 * (((-2.0d0) * im_m) + ((-0.3333333333333333d0) * (im_m ** 3.0d0)))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00072) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im_m));
	} else {
		tmp = 0.5 * ((-2.0 * im_m) + (-0.3333333333333333 * Math.pow(im_m, 3.0)));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.00072:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im_m))
	else:
		tmp = 0.5 * ((-2.0 * im_m) + (-0.3333333333333333 * math.pow(im_m, 3.0)))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.00072)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im_m)));
	else
		tmp = Float64(0.5 * Float64(Float64(-2.0 * im_m) + Float64(-0.3333333333333333 * (im_m ^ 3.0))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.00072)
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	else
		tmp = 0.5 * ((-2.0 * im_m) + (-0.3333333333333333 * (im_m ^ 3.0)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.00072], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * im$95$m), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 7.20000000000000045e-4

   1. Initial program 33.7%

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

      1. cos-neg 33.7%

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

      2. sub-neg 33.7%

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

      3. neg-sub0 33.7%

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

      4. remove-double-neg 33.7%

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

      5. remove-double-neg 33.7%

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

      6. sub0-neg 33.7%

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

      7. distribute-neg-in 33.7%

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

      8. +-commutative 33.7%

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

      9. sub-neg 33.7%

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

      10. associate-*l* 33.7%

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

      11. sub-neg 33.7%

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

      12. +-commutative 33.7%

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

      13. distribute-neg-in 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in im around 0 73.6%

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

   if 7.20000000000000045e-4 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 76.3%

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

   6. Taylor expanded in re around 0 57.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00072:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 5000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5000000000.0)
    (* 0.5 (* (cos re) (* -2.0 im_m)))
    (* (pow im_m 3.0) -0.16666666666666666))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5000000000.0) {
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * ((-2.0d0) * im_m))
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (-2.0 * im_m));
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5000000000.0:
		tmp = 0.5 * (math.cos(re) * (-2.0 * im_m))
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-2.0 * im_m)));
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5000000000.0)
		tmp = 0.5 * (cos(re) * (-2.0 * im_m));
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5e9

   1. Initial program 34.7%

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

      1. cos-neg 34.7%

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

      2. sub-neg 34.7%

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

      3. neg-sub0 34.7%

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

      4. remove-double-neg 34.7%

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

      5. remove-double-neg 34.7%

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

      6. sub0-neg 34.7%

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

      7. distribute-neg-in 34.7%

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

      8. +-commutative 34.7%

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

      9. sub-neg 34.7%

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

      10. associate-*l* 34.7%

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

      11. sub-neg 34.7%

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

      12. +-commutative 34.7%

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

      13. distribute-neg-in 34.7%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in im around 0 72.6%

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

   if 5e9 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 79.9%

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

   6. Taylor expanded in im around inf 79.9%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 60.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im_m\right)\\ \mathbf{else}:\\ \;\;\;\;{im_m}^{3} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.5)
    (* 0.5 (* -2.0 im_m))
    (* (pow im_m 3.0) -0.16666666666666666))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5) {
		tmp = 0.5 * (-2.0 * im_m);
	} else {
		tmp = pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.5d0) then
        tmp = 0.5d0 * ((-2.0d0) * im_m)
    else
        tmp = (im_m ** 3.0d0) * (-0.16666666666666666d0)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.5) {
		tmp = 0.5 * (-2.0 * im_m);
	} else {
		tmp = Math.pow(im_m, 3.0) * -0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.5:
		tmp = 0.5 * (-2.0 * im_m)
	else:
		tmp = math.pow(im_m, 3.0) * -0.16666666666666666
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.5)
		tmp = Float64(0.5 * Float64(-2.0 * im_m));
	else
		tmp = Float64((im_m ^ 3.0) * -0.16666666666666666);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.5)
		tmp = 0.5 * (-2.0 * im_m);
	else
		tmp = (im_m ^ 3.0) * -0.16666666666666666;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.5], N[(0.5 * N[(-2.0 * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.5

   1. Initial program 33.7%

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

      1. cos-neg 33.7%

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

      2. sub-neg 33.7%

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

      3. neg-sub0 33.7%

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

      4. remove-double-neg 33.7%

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

      5. remove-double-neg 33.7%

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

      6. sub0-neg 33.7%

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

      7. distribute-neg-in 33.7%

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

      8. +-commutative 33.7%

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

      9. sub-neg 33.7%

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

      10. associate-*l* 33.7%

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

      11. sub-neg 33.7%

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

      12. +-commutative 33.7%

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

      13. distribute-neg-in 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in im around 0 73.6%

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

   6. Taylor expanded in re around 0 40.3%

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

   if 2.5 < 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. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-sub0 100.0%

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

      4. remove-double-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub0-neg 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

   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 76.3%

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

   6. Taylor expanded in im around inf 76.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \cdot \cos re\right) \]

   7. Taylor expanded in re around 0 57.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.2% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. cos-neg 50.3%

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

   2. sub-neg 50.3%

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

   3. neg-sub0 50.3%

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

   4. remove-double-neg 50.3%

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

   5. remove-double-neg 50.3%

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

   6. sub0-neg 50.3%

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

   7. distribute-neg-in 50.3%

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

   8. +-commutative 50.3%

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

   9. sub-neg 50.3%

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

   10. associate-*l* 50.3%

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

   11. sub-neg 50.3%

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

   12. +-commutative 50.3%

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

   13. distribute-neg-in 50.3%

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

3. Simplified 50.3%

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

5. Taylor expanded in im around 0 56.7%

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

6. Taylor expanded in re around 0 31.4%

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

7. Final simplification 31.4%

   \[\leadsto 0.5 \cdot \left(-2 \cdot im\right) \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024026 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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