math.cos on complex, imaginary part

Percentage Accurate: 65.9% → 99.8%

Time: 11.7s

Alternatives: 17

Speedup: 2.4×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left(\left(-0.008333333333333333 \cdot {im\_m}^{5} + -0.0001984126984126984 \cdot {im\_m}^{7}\right) + -0.16666666666666666 \cdot {im\_m}^{3}\right) - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-5.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((((-0.008333333333333333d0) * (im_m ** 5.0d0)) + ((-0.0001984126984126984d0) * (im_m ** 7.0d0))) + ((-0.16666666666666666d0) * (im_m ** 3.0d0))) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((((-0.008333333333333333 * Math.pow(im_m, 5.0)) + (-0.0001984126984126984 * Math.pow(im_m, 7.0))) + (-0.16666666666666666 * Math.pow(im_m, 3.0))) - im_m);
	}
	return im_s * tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   if -5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.8%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. associate-+r+ 95.8%

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

      2. +-commutative 95.8%

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

      3. associate-*r* 95.8%

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

      4. associate-*r* 95.8%

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

      5. distribute-rgt-out 95.8%

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

      6. +-commutative 95.8%

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

      7. mul-1-neg 95.8%

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

      8. unsub-neg 95.8%

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

      9. associate-*r* 95.8%

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

      10. distribute-rgt-out-- 95.8%

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

      11. distribute-lft-out 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in re around inf 95.8%

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

4. Final simplification 97.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left(-0.008333333333333333 \cdot {im\_m}^{5} + -0.0001984126984126984 \cdot {im\_m}^{7}\right) + \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-5.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((-0.008333333333333333d0) * (im_m ** 5.0d0)) + ((-0.0001984126984126984d0) * (im_m ** 7.0d0))) + (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * (((-0.008333333333333333 * Math.pow(im_m, 5.0)) + (-0.0001984126984126984 * Math.pow(im_m, 7.0))) + ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m));
	}
	return im_s * tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   if -5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.8%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. associate-+r+ 95.8%

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

      2. +-commutative 95.8%

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

      3. associate-*r* 95.8%

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

      4. associate-*r* 95.8%

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

      5. distribute-rgt-out 95.8%

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

      6. +-commutative 95.8%

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

      7. mul-1-neg 95.8%

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

      8. unsub-neg 95.8%

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

      9. associate-*r* 95.8%

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

      10. distribute-rgt-out-- 95.8%

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

      11. distribute-lft-out 95.8%

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

   6. Simplified 95.8%

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

4. Final simplification 97.0%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.02d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   if -0.0200000000000000004 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 57.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.8%

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

   4. Taylor expanded in im around 0 92.6%

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

      1. +-commutative 92.6%

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

      2. mul-1-neg 92.6%

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

      3. unsub-neg 92.6%

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

      4. associate-*r* 92.6%

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

      5. distribute-rgt-out-- 92.6%

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

      6. *-commutative 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.02:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.115:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 0.115)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 1.1e+44)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* -0.0001984126984126984 (* (sin 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 <= 0.115) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 1.1e+44) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (sin(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 <= 0.115d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 1.1d+44) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(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 <= 0.115) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 1.1e+44) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(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 <= 0.115:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 1.1e+44:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(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 <= 0.115)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 1.1e+44)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(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 <= 0.115)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 1.1e+44)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = -0.0001984126984126984 * (sin(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, 0.115], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.1e+44], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.115000000000000005

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.8%

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

   4. Taylor expanded in im around 0 92.5%

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

      1. +-commutative 92.5%

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

      2. mul-1-neg 92.5%

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

      3. unsub-neg 92.5%

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

      4. associate-*r* 92.5%

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

      5. distribute-rgt-out-- 92.5%

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

      6. *-commutative 92.5%

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

   6. Simplified 92.5%

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

   if 0.115000000000000005 < im < 1.09999999999999998e44

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 80.0%

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

      1. associate-*r* 80.0%

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

      2. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   if 1.09999999999999998e44 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 100.0%

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

   4. Taylor expanded in im around inf 100.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.115:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 650:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 4.05 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im\_m \cdot re\right)\right)\\ \mathbf{elif}\;im\_m \leq 2.9 \cdot 10^{+197} \lor \neg \left(im\_m \leq 2.45 \cdot 10^{+230}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \end{array} \end{array} \]

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 650.0)
    (* (- im_m) (sin re))
    (if (<= im_m 4.05e+99)
      (log1p (expm1 (* im_m re)))
      (if (or (<= im_m 2.9e+197) (not (<= im_m 2.45e+230)))
        (* re (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
        (* im_m (- (* (pow re 3.0) 0.16666666666666666) re)))))))

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 <= 650.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 4.05e+99) {
		tmp = log1p(expm1((im_m * re)));
	} else if ((im_m <= 2.9e+197) || !(im_m <= 2.45e+230)) {
		tmp = re * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else {
		tmp = im_m * ((pow(re, 3.0) * 0.16666666666666666) - re);
	}
	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 <= 650.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 4.05e+99) {
		tmp = Math.log1p(Math.expm1((im_m * re)));
	} else if ((im_m <= 2.9e+197) || !(im_m <= 2.45e+230)) {
		tmp = re * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else {
		tmp = im_m * ((Math.pow(re, 3.0) * 0.16666666666666666) - re);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 650.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 4.05e+99:
		tmp = math.log1p(math.expm1((im_m * re)))
	elif (im_m <= 2.9e+197) or not (im_m <= 2.45e+230):
		tmp = re * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	else:
		tmp = im_m * ((math.pow(re, 3.0) * 0.16666666666666666) - re)
	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 <= 650.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 4.05e+99)
		tmp = log1p(expm1(Float64(im_m * re)));
	elseif ((im_m <= 2.9e+197) || !(im_m <= 2.45e+230))
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	else
		tmp = Float64(im_m * Float64(Float64((re ^ 3.0) * 0.16666666666666666) - re));
	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, 650.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.05e+99], N[Log[1 + N[(Exp[N[(im$95$m * re), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[im$95$m, 2.9e+197], N[Not[LessEqual[im$95$m, 2.45e+230]], $MachinePrecision]], N[(re * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 650

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

   if 650 < im < 4.05000000000000007e99

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 57.9%

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

      1. associate-*r* 57.9%

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

      2. *-commutative 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in im around 0 1.8%

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

      1. associate-*r* 1.8%

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

      2. *-commutative 1.8%

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

      3. associate-*l* 1.8%

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

      4. metadata-eval 1.8%

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

      5. *-commutative 1.8%

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

      6. associate-*r* 1.8%

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

      7. add-sqr-sqrt 0.9%

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

      8. sqrt-unprod 11.8%

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

      9. mul-1-neg 11.8%

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

      10. mul-1-neg 11.8%

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

      11. sqr-neg 11.8%

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

      12. sqrt-unprod 6.0%

          \[\leadsto \color{blue}{\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}} \]

      13. log1p-expm1-u 15.9%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}\right)\right)} \]

      14. add-sqr-sqrt 42.4%

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

      15. *-commutative 42.4%

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

   8. Applied egg-rr 42.4%

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

   if 4.05000000000000007e99 < im < 2.90000000000000002e197 or 2.44999999999999985e230 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 81.8%

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

      1. associate-*r* 81.8%

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

      2. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in im around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. associate-*r* 77.4%

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

      5. distribute-rgt-out-- 77.4%

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

      6. *-commutative 77.4%

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

   8. Simplified 77.4%

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

   if 2.90000000000000002e197 < im < 2.44999999999999985e230

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 5.0%

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

      1. associate-*r* 5.0%

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

      2. neg-mul-1 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0 31.6%

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

      1. +-commutative 31.6%

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

      2. mul-1-neg 31.6%

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

      3. unsub-neg 31.6%

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

      4. *-commutative 31.6%

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

      5. associate-*l* 31.6%

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

      6. distribute-lft-out-- 61.6%

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

   8. Simplified 61.6%

      \[\leadsto \color{blue}{im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 4.05 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+197} \lor \neg \left(im \leq 2.45 \cdot 10^{+230}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.5:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 5.5)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (* -0.0001984126984126984 (* (sin 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 <= 5.5) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else {
		tmp = -0.0001984126984126984 * (sin(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 <= 5.5d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else
        tmp = (-0.0001984126984126984d0) * (sin(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 <= 5.5) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(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 <= 5.5:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	else:
		tmp = -0.0001984126984126984 * (math.sin(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 <= 5.5)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(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 <= 5.5)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	else
		tmp = -0.0001984126984126984 * (sin(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, 5.5], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.5

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 95.8%

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

   4. Taylor expanded in im around 0 92.5%

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

      1. +-commutative 92.5%

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

      2. mul-1-neg 92.5%

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

      3. unsub-neg 92.5%

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

      4. associate-*r* 92.5%

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

      5. distribute-rgt-out-- 92.5%

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

      6. *-commutative 92.5%

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

   6. Simplified 92.5%

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

   if 5.5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.4%

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

   4. Taylor expanded in im around inf 93.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.2:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

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 4.2)
    (* (- im_m) (sin re))
    (* -0.0001984126984126984 (* (sin 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 <= 4.2) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -0.0001984126984126984 * (sin(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 <= 4.2d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(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 <= 4.2) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(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 <= 4.2:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(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 <= 4.2)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(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 <= 4.2)
		tmp = -im_m * sin(re);
	else
		tmp = -0.0001984126984126984 * (sin(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, 4.2], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.20000000000000018

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

   if 4.20000000000000018 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 93.4%

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

   4. Taylor expanded in im around inf 93.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4100:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+99} \lor \neg \left(im\_m \leq 2.05 \cdot 10^{+197}\right) \land im\_m \leq 2.45 \cdot 10^{+230}:\\ \;\;\;\;im\_m \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \]

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 4100.0)
    (* (- im_m) (sin re))
    (if (or (<= im_m 5.8e+99)
            (and (not (<= im_m 2.05e+197)) (<= im_m 2.45e+230)))
      (* im_m (- (* (pow re 3.0) 0.16666666666666666) re))
      (* re (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4100.0) {
		tmp = -im_m * sin(re);
	} else if ((im_m <= 5.8e+99) || (!(im_m <= 2.05e+197) && (im_m <= 2.45e+230))) {
		tmp = im_m * ((pow(re, 3.0) * 0.16666666666666666) - re);
	} else {
		tmp = re * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4100.0d0) then
        tmp = -im_m * sin(re)
    else if ((im_m <= 5.8d+99) .or. (.not. (im_m <= 2.05d+197)) .and. (im_m <= 2.45d+230)) then
        tmp = im_m * (((re ** 3.0d0) * 0.16666666666666666d0) - re)
    else
        tmp = re * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4100.0) {
		tmp = -im_m * Math.sin(re);
	} else if ((im_m <= 5.8e+99) || (!(im_m <= 2.05e+197) && (im_m <= 2.45e+230))) {
		tmp = im_m * ((Math.pow(re, 3.0) * 0.16666666666666666) - re);
	} else {
		tmp = re * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4100.0:
		tmp = -im_m * math.sin(re)
	elif (im_m <= 5.8e+99) or (not (im_m <= 2.05e+197) and (im_m <= 2.45e+230)):
		tmp = im_m * ((math.pow(re, 3.0) * 0.16666666666666666) - re)
	else:
		tmp = re * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4100.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif ((im_m <= 5.8e+99) || (!(im_m <= 2.05e+197) && (im_m <= 2.45e+230)))
		tmp = Float64(im_m * Float64(Float64((re ^ 3.0) * 0.16666666666666666) - re));
	else
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4100.0)
		tmp = -im_m * sin(re);
	elseif ((im_m <= 5.8e+99) || (~((im_m <= 2.05e+197)) && (im_m <= 2.45e+230)))
		tmp = im_m * (((re ^ 3.0) * 0.16666666666666666) - re);
	else
		tmp = re * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4100.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im$95$m, 5.8e+99], And[N[Not[LessEqual[im$95$m, 2.05e+197]], $MachinePrecision], LessEqual[im$95$m, 2.45e+230]]], N[(im$95$m * N[(N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4100

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

   if 4100 < im < 5.8000000000000004e99 or 2.05000000000000015e197 < im < 2.44999999999999985e230

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 3.8%

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

      1. associate-*r* 3.8%

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

      2. neg-mul-1 3.8%

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

   5. Simplified 3.8%

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

   6. Taylor expanded in re around 0 22.5%

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

      1. +-commutative 22.5%

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

      2. mul-1-neg 22.5%

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

      3. unsub-neg 22.5%

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

      4. *-commutative 22.5%

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

      5. associate-*l* 22.5%

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

      6. distribute-lft-out-- 43.2%

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

   8. Simplified 43.2%

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

   if 5.8000000000000004e99 < im < 2.05000000000000015e197 or 2.44999999999999985e230 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 81.8%

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

      1. associate-*r* 81.8%

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

      2. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in im around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. associate-*r* 77.4%

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

      5. distribute-rgt-out-- 77.4%

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

      6. *-commutative 77.4%

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

   8. Simplified 77.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4100:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+99} \lor \neg \left(im \leq 2.05 \cdot 10^{+197}\right) \land im \leq 2.45 \cdot 10^{+230}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 600:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.14 \cdot 10^{+195}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+244}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot -2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* im_m (- (* (pow re 3.0) 0.16666666666666666) re))))
   (*
    im_s
    (if (<= im_m 600.0)
      (* (- im_m) (sin re))
      (if (<= im_m 1.4e+162)
        t_0
        (if (<= im_m 1.14e+195)
          (* im_m (- re))
          (if (<= im_m 1.05e+244) t_0 (* (* 0.5 re) (* im_m -2.0)))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * ((pow(re, 3.0) * 0.16666666666666666) - re);
	double tmp;
	if (im_m <= 600.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.4e+162) {
		tmp = t_0;
	} else if (im_m <= 1.14e+195) {
		tmp = im_m * -re;
	} else if (im_m <= 1.05e+244) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im_m * (((re ** 3.0d0) * 0.16666666666666666d0) - re)
    if (im_m <= 600.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.4d+162) then
        tmp = t_0
    else if (im_m <= 1.14d+195) then
        tmp = im_m * -re
    else if (im_m <= 1.05d+244) then
        tmp = t_0
    else
        tmp = (0.5d0 * re) * (im_m * (-2.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = im_m * ((Math.pow(re, 3.0) * 0.16666666666666666) - re);
	double tmp;
	if (im_m <= 600.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.4e+162) {
		tmp = t_0;
	} else if (im_m <= 1.14e+195) {
		tmp = im_m * -re;
	} else if (im_m <= 1.05e+244) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = im_m * ((math.pow(re, 3.0) * 0.16666666666666666) - re)
	tmp = 0
	if im_m <= 600.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.4e+162:
		tmp = t_0
	elif im_m <= 1.14e+195:
		tmp = im_m * -re
	elif im_m <= 1.05e+244:
		tmp = t_0
	else:
		tmp = (0.5 * re) * (im_m * -2.0)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * Float64(Float64((re ^ 3.0) * 0.16666666666666666) - re))
	tmp = 0.0
	if (im_m <= 600.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.4e+162)
		tmp = t_0;
	elseif (im_m <= 1.14e+195)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 1.05e+244)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * -2.0));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = im_m * (((re ^ 3.0) * 0.16666666666666666) - re);
	tmp = 0.0;
	if (im_m <= 600.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.4e+162)
		tmp = t_0;
	elseif (im_m <= 1.14e+195)
		tmp = im_m * -re;
	elseif (im_m <= 1.05e+244)
		tmp = t_0;
	else
		tmp = (0.5 * re) * (im_m * -2.0);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 600.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+162], t$95$0, If[LessEqual[im$95$m, 1.14e+195], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 1.05e+244], t$95$0, N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if im < 600

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

   if 600 < im < 1.39999999999999995e162 or 1.13999999999999997e195 < im < 1.05e244

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 3.7%

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

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

   5. Simplified 3.7%

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

   6. Taylor expanded in re around 0 17.0%

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

      1. +-commutative 17.0%

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

      2. mul-1-neg 17.0%

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

      3. unsub-neg 17.0%

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

      4. *-commutative 17.0%

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

      5. associate-*l* 17.0%

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

      6. distribute-lft-out-- 34.0%

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

   8. Simplified 34.0%

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

   if 1.39999999999999995e162 < im < 1.13999999999999997e195

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around 0 51.8%

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

      1. associate-*r* 51.8%

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

      2. neg-mul-1 51.8%

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

   8. Simplified 51.8%

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

   if 1.05e244 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0 45.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+162}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{elif}\;im \leq 1.14 \cdot 10^{+195}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+244}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 650:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 8.8 \cdot 10^{+162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{elif}\;im\_m \leq 2.1 \cdot 10^{+243}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot -2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (* im_m (pow re 3.0)))))
   (*
    im_s
    (if (<= im_m 650.0)
      (* (- im_m) (sin re))
      (if (<= im_m 8.8e+162)
        t_0
        (if (<= im_m 4.3e+196)
          (* im_m (- re))
          (if (<= im_m 2.1e+243) t_0 (* (* 0.5 re) (* im_m -2.0)))))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.16666666666666666 * (im_m * pow(re, 3.0));
	double tmp;
	if (im_m <= 650.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 8.8e+162) {
		tmp = t_0;
	} else if (im_m <= 4.3e+196) {
		tmp = im_m * -re;
	} else if (im_m <= 2.1e+243) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    if (im_m <= 650.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 8.8d+162) then
        tmp = t_0
    else if (im_m <= 4.3d+196) then
        tmp = im_m * -re
    else if (im_m <= 2.1d+243) then
        tmp = t_0
    else
        tmp = (0.5d0 * re) * (im_m * (-2.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	double tmp;
	if (im_m <= 650.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 8.8e+162) {
		tmp = t_0;
	} else if (im_m <= 4.3e+196) {
		tmp = im_m * -re;
	} else if (im_m <= 2.1e+243) {
		tmp = t_0;
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	tmp = 0
	if im_m <= 650.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 8.8e+162:
		tmp = t_0
	elif im_m <= 4.3e+196:
		tmp = im_m * -re
	elif im_m <= 2.1e+243:
		tmp = t_0
	else:
		tmp = (0.5 * re) * (im_m * -2.0)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)))
	tmp = 0.0
	if (im_m <= 650.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 8.8e+162)
		tmp = t_0;
	elseif (im_m <= 4.3e+196)
		tmp = Float64(im_m * Float64(-re));
	elseif (im_m <= 2.1e+243)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * -2.0));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.16666666666666666 * (im_m * (re ^ 3.0));
	tmp = 0.0;
	if (im_m <= 650.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 8.8e+162)
		tmp = t_0;
	elseif (im_m <= 4.3e+196)
		tmp = im_m * -re;
	elseif (im_m <= 2.1e+243)
		tmp = t_0;
	else
		tmp = (0.5 * re) * (im_m * -2.0);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 650.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 8.8e+162], t$95$0, If[LessEqual[im$95$m, 4.3e+196], N[(im$95$m * (-re)), $MachinePrecision], If[LessEqual[im$95$m, 2.1e+243], t$95$0, N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if im < 650

   1. Initial program 57.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

   if 650 < im < 8.8000000000000007e162 or 4.30000000000000012e196 < im < 2.0999999999999999e243

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 3.7%

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

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

   5. Simplified 3.7%

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

   6. Taylor expanded in re around 0 17.0%

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

   7. Taylor expanded in re around inf 33.2%

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

   if 8.8000000000000007e162 < im < 4.30000000000000012e196

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around 0 51.8%

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

      1. associate-*r* 51.8%

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

      2. neg-mul-1 51.8%

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

   8. Simplified 51.8%

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

   if 2.0999999999999999e243 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0 45.5%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 8.8 \cdot 10^{+162}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+196}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+243}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.75 \cdot 10^{+106}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.75e+106) (* (- im_m) (sin re)) (* (* 0.5 re) (* im_m -2.0)))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.75e+106) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.75d+106) then
        tmp = -im_m * sin(re)
    else
        tmp = (0.5d0 * re) * (im_m * (-2.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 1.75e+106) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (0.5 * re) * (im_m * -2.0);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 1.75e+106:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (0.5 * re) * (im_m * -2.0)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 1.75e+106)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * -2.0));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 1.75e+106)
		tmp = -im_m * sin(re);
	else
		tmp = (0.5 * re) * (im_m * -2.0);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 1.75e+106], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.7499999999999999e106

   1. Initial program 61.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. neg-mul-1 63.4%

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

   5. Simplified 63.4%

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

   if 1.7499999999999999e106 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in im around 0 27.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{+106}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.5% accurate, 21.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+167} \lor \neg \left(re \leq 6.4 \cdot 10^{+291}\right):\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= re 1.08e+167) (not (<= re 6.4e+291)))
    (* im_m (- re))
    (* im_m re))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((re <= 1.08e+167) || !(re <= 6.4e+291)) {
		tmp = im_m * -re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((re <= 1.08d+167) .or. (.not. (re <= 6.4d+291))) then
        tmp = im_m * -re
    else
        tmp = im_m * re
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((re <= 1.08e+167) || !(re <= 6.4e+291)) {
		tmp = im_m * -re;
	} else {
		tmp = im_m * re;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (re <= 1.08e+167) or not (re <= 6.4e+291):
		tmp = im_m * -re
	else:
		tmp = im_m * re
	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 ((re <= 1.08e+167) || !(re <= 6.4e+291))
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(im_m * re);
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((re <= 1.08e+167) || ~((re <= 6.4e+291)))
		tmp = im_m * -re;
	else
		tmp = im_m * re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[Or[LessEqual[re, 1.08e+167], N[Not[LessEqual[re, 6.4e+291]], $MachinePrecision]], N[(im$95$m * (-re)), $MachinePrecision], N[(im$95$m * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.08000000000000005e167 or 6.4000000000000004e291 < re

   1. Initial program 71.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 52.1%

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

      1. associate-*r* 52.1%

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

      2. neg-mul-1 52.1%

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

   5. Simplified 52.1%

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

   6. Taylor expanded in re around 0 36.4%

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

      1. associate-*r* 36.4%

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

      2. neg-mul-1 36.4%

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

   8. Simplified 36.4%

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

   if 1.08000000000000005e167 < re < 6.4000000000000004e291

   1. Initial program 57.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 23.4%

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

      1. associate-*r* 23.4%

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

      2. *-commutative 23.4%

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

   5. Simplified 23.4%

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

   6. Taylor expanded in im around 0 15.7%

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

      1. associate-*r* 15.7%

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

      2. *-commutative 15.7%

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

      3. associate-*l* 15.7%

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

      4. metadata-eval 15.7%

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

      5. *-commutative 15.7%

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

      6. associate-*r* 15.7%

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

      7. add-sqr-sqrt 4.6%

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

      8. sqrt-unprod 25.9%

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

      9. mul-1-neg 25.9%

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

      10. mul-1-neg 25.9%

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

      11. sqr-neg 25.9%

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

      12. sqrt-unprod 14.9%

          \[\leadsto \color{blue}{\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}} \]

      13. add-sqr-sqrt 29.6%

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

      14. expm1-log1p-u 15.0%

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

      15. expm1-udef 15.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im \cdot re\right)} - 1} \]

      16. *-commutative 15.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{re \cdot im}\right)} - 1 \]

   8. Applied egg-rr 15.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re \cdot im\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 15.0%

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

      2. expm1-log1p 29.6%

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

   10. Simplified 29.6%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+167} \lor \neg \left(re \leq 6.4 \cdot 10^{+291}\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.5% accurate, 22.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+167}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot -2\right)\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+291}:\\ \;\;\;\;im\_m \cdot re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.08e+167)
    (* (* 0.5 re) (* im_m -2.0))
    (if (<= re 6.4e+291) (* im_m re) (* im_m (- re))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.08e+167) {
		tmp = (0.5 * re) * (im_m * -2.0);
	} else if (re <= 6.4e+291) {
		tmp = im_m * re;
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.08d+167) then
        tmp = (0.5d0 * re) * (im_m * (-2.0d0))
    else if (re <= 6.4d+291) then
        tmp = im_m * re
    else
        tmp = im_m * -re
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.08e+167) {
		tmp = (0.5 * re) * (im_m * -2.0);
	} else if (re <= 6.4e+291) {
		tmp = im_m * re;
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.08e+167:
		tmp = (0.5 * re) * (im_m * -2.0)
	elif re <= 6.4e+291:
		tmp = im_m * re
	else:
		tmp = im_m * -re
	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 (re <= 1.08e+167)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * -2.0));
	elseif (re <= 6.4e+291)
		tmp = Float64(im_m * re);
	else
		tmp = Float64(im_m * Float64(-re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.08e+167)
		tmp = (0.5 * re) * (im_m * -2.0);
	elseif (re <= 6.4e+291)
		tmp = im_m * re;
	else
		tmp = im_m * -re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.08e+167], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e+291], N[(im$95$m * re), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if re < 1.08000000000000005e167

   1. Initial program 70.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 56.4%

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

      1. associate-*r* 56.4%

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

      2. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in im around 0 36.7%

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

   if 1.08000000000000005e167 < re < 6.4000000000000004e291

   1. Initial program 57.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0 23.4%

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

      1. associate-*r* 23.4%

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

      2. *-commutative 23.4%

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

   5. Simplified 23.4%

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

   6. Taylor expanded in im around 0 15.7%

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

      1. associate-*r* 15.7%

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

      2. *-commutative 15.7%

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

      3. associate-*l* 15.7%

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

      4. metadata-eval 15.7%

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

      5. *-commutative 15.7%

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

      6. associate-*r* 15.7%

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

      7. add-sqr-sqrt 4.6%

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

      8. sqrt-unprod 25.9%

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

      9. mul-1-neg 25.9%

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

      10. mul-1-neg 25.9%

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

      11. sqr-neg 25.9%

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

      12. sqrt-unprod 14.9%

          \[\leadsto \color{blue}{\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}} \]

      13. add-sqr-sqrt 29.6%

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

      14. expm1-log1p-u 15.0%

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

      15. expm1-udef 15.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im \cdot re\right)} - 1} \]

      16. *-commutative 15.7%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{re \cdot im}\right)} - 1 \]

   8. Applied egg-rr 15.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re \cdot im\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 15.0%

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

      2. expm1-log1p 29.6%

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

   10. Simplified 29.6%

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

   if 6.4000000000000004e291 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0 4.4%

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

      1. associate-*r* 4.4%

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

      2. neg-mul-1 4.4%

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

   5. Simplified 4.4%

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

   6. Taylor expanded in re around 0 50.0%

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

      1. associate-*r* 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.08 \cdot 10^{+167}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{+291}:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 21.1% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0 52.6%

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

   1. associate-*r* 52.6%

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

   2. *-commutative 52.6%

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

5. Simplified 52.6%

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

6. Taylor expanded in im around 0 34.4%

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

   1. associate-*r* 34.4%

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

   2. *-commutative 34.4%

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

   3. associate-*l* 34.1%

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

   4. metadata-eval 34.1%

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

   5. *-commutative 34.1%

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

   6. associate-*r* 34.1%

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

   7. add-sqr-sqrt 27.8%

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

   8. sqrt-unprod 33.1%

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

   9. mul-1-neg 33.1%

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

   10. mul-1-neg 33.1%

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

   11. sqr-neg 33.1%

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

   12. sqrt-unprod 19.8%

       \[\leadsto \color{blue}{\sqrt{im \cdot re} \cdot \sqrt{im \cdot re}} \]

   13. add-sqr-sqrt 25.2%

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

   14. expm1-log1p-u 20.5%

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

   15. expm1-udef 20.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(im \cdot re\right)} - 1} \]

   16. *-commutative 20.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{re \cdot im}\right)} - 1 \]

8. Applied egg-rr 20.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re \cdot im\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 20.5%

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

   2. expm1-log1p 25.2%

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

10. Simplified 25.2%

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

11. Final simplification 25.2%

    \[\leadsto im \cdot re \]
12. Add Preprocessing

Alternative 15: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -8 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -8.0))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -8.0;
}

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 * (-8.0d0)
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 * -8.0;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -8.0

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -8.0)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -8.0;
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 * -8.0), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 95.1%

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

5. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-8} \]

6. Final simplification 2.6%

   \[\leadsto -8 \]
7. Add Preprocessing

Alternative 16: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -6.248825220858479 \cdot 10^{-11} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -6.248825220858479e-11))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -6.248825220858479e-11;
}

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 * (-6.248825220858479d-11)
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 * -6.248825220858479e-11;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -6.248825220858479e-11

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -6.248825220858479e-11)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -6.248825220858479e-11;
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 * -6.248825220858479e-11), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 95.1%

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

5. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-6.248825220858479 \cdot 10^{-11}} \]

6. Final simplification 2.7%

   \[\leadsto -6.248825220858479 \cdot 10^{-11} \]
7. Add Preprocessing

Alternative 17: 15.5% accurate, 308.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 95.1%

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

5. Applied egg-rr 18.7%

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

6. Final simplification 18.7%

   \[\leadsto 0 \]
7. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-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 = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024031 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

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

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))