math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.3%

Time: 10.3s

Alternatives: 13

Speedup: 2.8×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 65.7% 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: 99.3% accurate, 0.6× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 (* 0.5 (sin re)))
      (* im_m (* (sin re) (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = im_m * (Math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = im_m * (math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(sin(re) * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 82.2%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 82.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. associate-*r* 82.2%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

      3. distribute-rgt-out 82.2%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 82.2%

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

   7. Applied egg-rr 82.2%

      \[\leadsto im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(im \cdot im\right)} + -1\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 105:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \mathbf{elif}\;im\_m \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 105.0)
    (* im_m (* (sin re) (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))
    (if (<= im_m 3.3e+100)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (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 tmp;
	if (im_m <= 105.0) {
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else if (im_m <= 3.3e+100) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} 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) :: tmp
    if (im_m <= 105.0d0) then
        tmp = im_m * (sin(re) * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    else if (im_m <= 3.3d+100) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    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 tmp;
	if (im_m <= 105.0) {
		tmp = im_m * (Math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else if (im_m <= 3.3e+100) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} 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):
	tmp = 0
	if im_m <= 105.0:
		tmp = im_m * (math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	elif im_m <= 3.3e+100:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	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)
	tmp = 0.0
	if (im_m <= 105.0)
		tmp = Float64(im_m * Float64(sin(re) * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	elseif (im_m <= 3.3e+100)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	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)
	tmp = 0.0;
	if (im_m <= 105.0)
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	elseif (im_m <= 3.3e+100)
		tmp = 8.0 * (27.0 - exp(im_m));
	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_] := N[(im$95$s * If[LessEqual[im$95$m, 105.0], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.3e+100], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $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 3 regimes

2. if im < 105

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 82.2%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 82.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. associate-*r* 82.2%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

      3. distribute-rgt-out 82.2%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 82.2%

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

   7. Applied egg-rr 82.2%

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

   if 105 < im < 3.3000000000000001e100

   1. Initial program 100.0%

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

   4. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 37.5%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 3.3000000000000001e100 < 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 89.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 89.3%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. distribute-lft-in 89.3%

         \[\leadsto \color{blue}{im \cdot \left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      3. associate-*r* 89.3%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      4. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} + im \cdot \left(-1 \cdot \sin re\right) \]

      5. associate-*r* 98.3%

         \[\leadsto \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(im \cdot -1\right) \cdot \sin re} \]

      6. *-commutative 98.3%

         \[\leadsto \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re + \color{blue}{\left(-1 \cdot im\right)} \cdot \sin re \]

      7. distribute-rgt-out 98.3%

         \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) + -1 \cdot im\right)} \]

      8. neg-mul-1 98.3%

         \[\leadsto \sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) + \color{blue}{\left(-im\right)}\right) \]

      9. unsub-neg 98.3%

         \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      10. *-commutative 98.3%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      11. associate-*r* 98.3%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

      12. unpow2 98.3%

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

      13. cube-unmult 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 105:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 190:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im\_m \cdot im\_m}{-im\_m}\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 190.0)
    (* im_m (* (sin re) (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))
    (if (<= im_m 1.35e+154)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (sin re) (/ (* im_m im_m) (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 190.0) {
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else if (im_m <= 1.35e+154) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = sin(re) * ((im_m * im_m) / -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 <= 190.0d0) then
        tmp = im_m * (sin(re) * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    else if (im_m <= 1.35d+154) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = sin(re) * ((im_m * im_m) / -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 <= 190.0) {
		tmp = im_m * (Math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else if (im_m <= 1.35e+154) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * ((im_m * im_m) / -im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 190.0:
		tmp = im_m * (math.sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	elif im_m <= 1.35e+154:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.sin(re) * ((im_m * im_m) / -im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 190.0)
		tmp = Float64(im_m * Float64(sin(re) * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	elseif (im_m <= 1.35e+154)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64(im_m * im_m) / Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 190.0)
		tmp = im_m * (sin(re) * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	elseif (im_m <= 1.35e+154)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = sin(re) * ((im_m * im_m) / -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, 190.0], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.35e+154], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-im$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 190

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 82.2%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 82.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. associate-*r* 82.2%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

      3. distribute-rgt-out 82.2%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 82.2%

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

   7. Applied egg-rr 82.2%

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

   if 190 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   4. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 41.9%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 1.35000000000000003e154 < 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 5.2%

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

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

   5. Simplified 5.2%

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

      1. neg-sub0 5.2%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \cdot \sin re \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0} - im \cdot im}{0 + im} \cdot \sin re \]

      4. unpow2 100.0%

         \[\leadsto \frac{0 - \color{blue}{{im}^{2}}}{0 + im} \cdot \sin re \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      6. sqrt-prod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im \cdot im}}} \cdot \sin re \]

      7. sqr-neg 0.0%

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

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      9. add-sqr-sqrt 0.0%

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

      10. sub-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{0 - im}} \cdot \sin re \]

      11. neg-sub0 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{-im}} \cdot \sin re \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      13. sqrt-unprod 0.0%

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

      14. sqr-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\sqrt{\color{blue}{im \cdot im}}} \cdot \sin re \]

      15. sqrt-prod 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{im}} \cdot \sin re \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0 - {im}^{2}}{im}} \cdot \sin re \]
   8. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{-{im}^{2}}}{im} \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-{im}^{2}}{im}} \cdot \sin re \]
   10. Step-by-step derivation

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 190:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im \cdot im}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 195:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im\_m \cdot im\_m}{-im\_m}\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 195.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.35e+154)
      (* 8.0 (- 27.0 (exp im_m)))
      (* (sin re) (/ (* im_m im_m) (- im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 195.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.35e+154) {
		tmp = 8.0 * (27.0 - exp(im_m));
	} else {
		tmp = sin(re) * ((im_m * im_m) / -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 <= 195.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.35d+154) then
        tmp = 8.0d0 * (27.0d0 - exp(im_m))
    else
        tmp = sin(re) * ((im_m * im_m) / -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 <= 195.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.35e+154) {
		tmp = 8.0 * (27.0 - Math.exp(im_m));
	} else {
		tmp = Math.sin(re) * ((im_m * im_m) / -im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 195.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.35e+154:
		tmp = 8.0 * (27.0 - math.exp(im_m))
	else:
		tmp = math.sin(re) * ((im_m * im_m) / -im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 195.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.35e+154)
		tmp = Float64(8.0 * Float64(27.0 - exp(im_m)));
	else
		tmp = Float64(sin(re) * Float64(Float64(im_m * im_m) / Float64(-im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 195.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.35e+154)
		tmp = 8.0 * (27.0 - exp(im_m));
	else
		tmp = sin(re) * ((im_m * im_m) / -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, 195.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.35e+154], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] / (-im$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 195

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 195 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   4. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 41.9%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 1.35000000000000003e154 < 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 5.2%

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

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

   5. Simplified 5.2%

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

      1. neg-sub0 5.2%

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

      2. flip-- 100.0%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - im \cdot im}{0 + im}} \cdot \sin re \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0} - im \cdot im}{0 + im} \cdot \sin re \]

      4. unpow2 100.0%

         \[\leadsto \frac{0 - \color{blue}{{im}^{2}}}{0 + im} \cdot \sin re \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      6. sqrt-prod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{im \cdot im}}} \cdot \sin re \]

      7. sqr-neg 0.0%

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

      8. sqrt-unprod 0.0%

         \[\leadsto \frac{0 - {im}^{2}}{0 + \color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      9. add-sqr-sqrt 0.0%

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

      10. sub-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{0 - im}} \cdot \sin re \]

      11. neg-sub0 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{-im}} \cdot \sin re \]

      12. add-sqr-sqrt 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} \cdot \sin re \]

      13. sqrt-unprod 0.0%

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

      14. sqr-neg 0.0%

          \[\leadsto \frac{0 - {im}^{2}}{\sqrt{\color{blue}{im \cdot im}}} \cdot \sin re \]

      15. sqrt-prod 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} \cdot \sin re \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \frac{0 - {im}^{2}}{\color{blue}{im}} \cdot \sin re \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{0 - {im}^{2}}{im}} \cdot \sin re \]
   8. Step-by-step derivation

      1. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{-{im}^{2}}}{im} \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-{im}^{2}}{im}} \cdot \sin re \]
   10. Step-by-step derivation

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;8 \cdot \left(27 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \frac{im \cdot im}{-im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 17000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(27 - e^{im\_m}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 17000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 5e+102)
      (* (- 27.0 (exp im_m)) -2.0)
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

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 <= 17000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 5e+102) {
		tmp = (27.0 - exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 17000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 5d+102) then
        tmp = (27.0d0 - exp(im_m)) * (-2.0d0)
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 17000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 5e+102) {
		tmp = (27.0 - Math.exp(im_m)) * -2.0;
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 17000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 5e+102:
		tmp = (27.0 - math.exp(im_m)) * -2.0
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 17000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 5e+102)
		tmp = Float64(Float64(27.0 - exp(im_m)) * -2.0);
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 17000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 5e+102)
		tmp = (27.0 - exp(im_m)) * -2.0;
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 17000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5e+102], N[(N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 17000

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 17000 < im < 5e102

   1. Initial program 100.0%

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

   4. Applied egg-rr 64.7%

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

   6. Applied egg-rr 64.7%

      \[\leadsto -2 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   if 5e102 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 49.0%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 49.0%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 17000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(27 - e^{im}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 105:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(27 - e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 105.0) (* (- im_m) (sin re)) (* 8.0 (- 27.0 (exp 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 <= 105.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = 8.0 * (27.0 - exp(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 <= 105.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = 8.0d0 * (27.0d0 - exp(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 <= 105.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = 8.0 * (27.0 - Math.exp(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 <= 105.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = 8.0 * (27.0 - math.exp(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 <= 105.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(8.0 * Float64(27.0 - exp(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 <= 105.0)
		tmp = -im_m * sin(re);
	else
		tmp = 8.0 * (27.0 - exp(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, 105.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(27.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 105

   1. Initial program 54.5%

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

   3. Taylor expanded in im around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 105 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 45.6%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 6 \cdot 10^{+96}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4e-5)
    (* (- im_m) (sin re))
    (if (<= im_m 6e+96)
      (* im_m (* re (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))
      (+
       208.0
       (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0)))))))

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 <= 4e-5) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 6e+96) {
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 4d-5) then
        tmp = -im_m * sin(re)
    else if (im_m <= 6d+96) then
        tmp = im_m * (re * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 4e-5) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 6e+96) {
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 4e-5:
		tmp = -im_m * math.sin(re)
	elif im_m <= 6e+96:
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 4e-5)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 6e+96)
		tmp = Float64(im_m * Float64(re * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 4e-5)
		tmp = -im_m * sin(re);
	elseif (im_m <= 6e+96)
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 4e-5], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6e+96], N[(im$95$m * N[(re * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.00000000000000033e-5

   1. Initial program 54.4%

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

   3. Taylor expanded in im around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 4.00000000000000033e-5 < im < 6.0000000000000001e96

   1. Initial program 99.1%

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

   3. Taylor expanded in im around 0 10.2%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 10.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. associate-*r* 10.2%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

      3. distribute-rgt-out 10.2%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

   5. Simplified 10.2%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 10.2%

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

   7. Applied egg-rr 10.2%

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

   8. Taylor expanded in re around 0 30.4%

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

   if 6.0000000000000001e96 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 48.1%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 48.1%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+96}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6 \cdot 10^{+96}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6e+96)
    (* im_m (* re (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))
    (+ 208.0 (* im_m (- (* im_m (- (* im_m -1.3333333333333333) 4.0)) 8.0))))))

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 <= 6e+96) {
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 6d+96) then
        tmp = im_m * (re * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0)))
    else
        tmp = 208.0d0 + (im_m * ((im_m * ((im_m * (-1.3333333333333333d0)) - 4.0d0)) - 8.0d0))
    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 <= 6e+96) {
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	} else {
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	}
	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 <= 6e+96:
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0))
	else:
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0))
	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 <= 6e+96)
		tmp = Float64(im_m * Float64(re * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0)));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -1.3333333333333333) - 4.0)) - 8.0)));
	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 <= 6e+96)
		tmp = im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0));
	else
		tmp = 208.0 + (im_m * ((im_m * ((im_m * -1.3333333333333333) - 4.0)) - 8.0));
	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, 6e+96], N[(im$95$m * N[(re * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -1.3333333333333333), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] - 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.0000000000000001e96

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 76.1%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 76.1%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. associate-*r* 76.1%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

      3. distribute-rgt-out 76.1%

         \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

   5. Simplified 76.1%

      \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 76.1%

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

   7. Applied egg-rr 76.1%

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

   8. Taylor expanded in re around 0 47.2%

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

   if 6.0000000000000001e96 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 48.1%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 48.1%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 48.1%

      \[\leadsto \color{blue}{208 + im \cdot \left(im \cdot \left(-1.3333333333333333 \cdot im - 4\right) - 8\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6 \cdot 10^{+96}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot \left(im \cdot -1.3333333333333333 - 4\right) - 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.0% accurate, 25.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im\_m \cdot \left(im\_m \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.15e+146) (* im_m (- re)) (+ 208.0 (* im_m (* im_m -4.0))))))

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 <= 1.15e+146) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * (im_m * -4.0));
	}
	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 <= 1.15d+146) then
        tmp = im_m * -re
    else
        tmp = 208.0d0 + (im_m * (im_m * (-4.0d0)))
    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 <= 1.15e+146) {
		tmp = im_m * -re;
	} else {
		tmp = 208.0 + (im_m * (im_m * -4.0));
	}
	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 <= 1.15e+146:
		tmp = im_m * -re
	else:
		tmp = 208.0 + (im_m * (im_m * -4.0))
	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 <= 1.15e+146)
		tmp = Float64(im_m * Float64(-re));
	else
		tmp = Float64(208.0 + Float64(im_m * Float64(im_m * -4.0)));
	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 <= 1.15e+146)
		tmp = im_m * -re;
	else
		tmp = 208.0 + (im_m * (im_m * -4.0));
	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, 1.15e+146], N[(im$95$m * (-re)), $MachinePrecision], N[(208.0 + N[(im$95$m * N[(im$95$m * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.15e146

   1. Initial program 60.6%

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

   3. Taylor expanded in im around 0 58.3%

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

      1. associate-*r* 58.3%

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

      2. neg-mul-1 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in re around 0 34.0%

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

   if 1.15e146 < im

   1. Initial program 100.0%

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

   4. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

   6. Applied egg-rr 46.2%

      \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

   7. Taylor expanded in im around 0 46.2%

      \[\leadsto \color{blue}{208 + im \cdot \left(-4 \cdot im - 8\right)} \]

   8. Taylor expanded in im around inf 46.2%

      \[\leadsto 208 + im \cdot \color{blue}{\left(-4 \cdot im\right)} \]
   9. Step-by-step derivation

      1. *-commutative 46.2%

         \[\leadsto 208 + im \cdot \color{blue}{\left(im \cdot -4\right)} \]

   10. Simplified 46.2%

       \[\leadsto 208 + im \cdot \color{blue}{\left(im \cdot -4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;208 + im \cdot \left(im \cdot -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* re (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (re * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(re * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in im around 0 78.8%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + -0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 78.8%

      \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

   2. associate-*r* 78.8%

      \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re} + -1 \cdot \sin re\right) \]

   3. distribute-rgt-out 78.8%

      \[\leadsto im \cdot \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]

5. Simplified 78.8%

   \[\leadsto \color{blue}{im \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{2} + -1\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 78.8%

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

7. Applied egg-rr 78.8%

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

8. Taylor expanded in re around 0 53.4%

   \[\leadsto im \cdot \left(\color{blue}{re} \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right) \]
9. Add Preprocessing

Alternative 11: 33.1% accurate, 77.0× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m (- 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 * Float64(-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 66.6%

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

3. Taylor expanded in im around 0 50.2%

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

   1. associate-*r* 50.2%

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

   2. neg-mul-1 50.2%

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

5. Simplified 50.2%

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

6. Taylor expanded in re around 0 32.0%

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

7. Final simplification 32.0%

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

Alternative 12: 20.2% 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 #s(literal 1 binary64) 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 66.6%

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

3. Taylor expanded in im around 0 50.2%

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

   1. associate-*r* 50.2%

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

   2. neg-mul-1 50.2%

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

5. Simplified 50.2%

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

6. Taylor expanded in re around 0 32.0%

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

   1. neg-sub0 32.0%

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

   2. sub-neg 32.0%

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

   3. add-sqr-sqrt 12.5%

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

   4. sqrt-unprod 27.0%

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

   5. sqr-neg 27.0%

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

   6. sqrt-prod 8.3%

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

   7. add-sqr-sqrt 17.6%

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

8. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{\left(0 + im\right)} \cdot re \]
9. Step-by-step derivation

   1. +-lft-identity 17.6%

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

10. Simplified 17.6%

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

Alternative 13: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 208 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 208.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 208.0;
}

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 * 208.0d0
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 * 208.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 208.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 208.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 208.0;
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 * 208.0), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

4. Applied egg-rr 37.8%

   \[\leadsto \color{blue}{8} \cdot \left(e^{-im} - e^{im}\right) \]

6. Applied egg-rr 14.1%

   \[\leadsto 8 \cdot \left(\color{blue}{27} - e^{im}\right) \]

7. Taylor expanded in im around 0 2.5%

   \[\leadsto \color{blue}{208} \]
8. Add Preprocessing

Developer Target 1: 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 2024165 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

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