math.cos on complex, imaginary part

Percentage Accurate: 66.6% → 99.4%

Time: 9.5s

Alternatives: 13

Speedup: 2.8×

• 50.1% 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: 66.6% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im_m} - e^{im_m}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \end{array} \end{array} \end{array} \]

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 -2.0) (* t_0 (* 0.5 (sin re))) (* im_m (- (sin re)))))))

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 <= -2.0) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im_m * -sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-2.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = im_m * -sin(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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = im_m * -Math.sin(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):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -2.0:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = im_m * -math.sin(re)
	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 <= -2.0)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(-sin(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)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = im_m * -sin(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_] := 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, -2.0], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 49.2%

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

   3. Taylor expanded in im around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. neg-mul-1 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 79.6%

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

Alternative 2: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 50000 \lor \neg \left(im_m \leq 1.3 \cdot 10^{+101}\right):\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (or (<= im_m 50000.0) (not (<= im_m 1.3e+101)))
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (* im_m (- (* 0.16666666666666666 (pow re 3.0)) 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 <= 50000.0) || !(im_m <= 1.3e+101)) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else {
		tmp = im_m * ((0.16666666666666666 * pow(re, 3.0)) - 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 ((im_m <= 50000.0d0) .or. (.not. (im_m <= 1.3d+101))) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else
        tmp = im_m * ((0.16666666666666666d0 * (re ** 3.0d0)) - 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 ((im_m <= 50000.0) || !(im_m <= 1.3e+101)) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else {
		tmp = im_m * ((0.16666666666666666 * Math.pow(re, 3.0)) - 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 <= 50000.0) or not (im_m <= 1.3e+101):
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	else:
		tmp = im_m * ((0.16666666666666666 * math.pow(re, 3.0)) - 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 <= 50000.0) || !(im_m <= 1.3e+101))
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	else
		tmp = Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - 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 ((im_m <= 50000.0) || ~((im_m <= 1.3e+101)))
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	else
		tmp = im_m * ((0.16666666666666666 * (re ^ 3.0)) - 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[im$95$m, 50000.0], N[Not[LessEqual[im$95$m, 1.3e+101]], $MachinePrecision]], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5e4 or 1.3e101 < im

   1. Initial program 59.6%

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

   3. Taylor expanded in im around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. neg-mul-1 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. *-commutative 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in re around inf 91.4%

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

   if 5e4 < im < 1.3e101

   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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 22.8%

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

      1. +-commutative 22.8%

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

      2. mul-1-neg 22.8%

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

      3. unsub-neg 22.8%

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

      4. *-commutative 22.8%

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

      5. associate-*r* 22.8%

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

      6. *-commutative 22.8%

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

      7. distribute-rgt-out-- 31.5%

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

   8. Simplified 31.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 50000 \lor \neg \left(im \leq 1.3 \cdot 10^{+101}\right):\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im_m}^{3}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 330:\\ \;\;\;\;\sin re \cdot \left(t_0 - im_m\right)\\ \mathbf{elif}\;im_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im_m} - e^{im_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \end{array} \]

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 (pow im_m 3.0))))
   (*
    im_s
    (if (<= im_m 330.0)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 5.5e+102)
        (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
        (* (sin re) t_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 * pow(im_m, 3.0);
	double tmp;
	if (im_m <= 330.0) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.5e+102) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = sin(re) * t_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 ** 3.0d0)
    if (im_m <= 330.0d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 5.5d+102) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = sin(re) * t_0
    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 * Math.pow(im_m, 3.0);
	double tmp;
	if (im_m <= 330.0) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 5.5e+102) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * t_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 * math.pow(im_m, 3.0)
	tmp = 0
	if im_m <= 330.0:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 5.5e+102:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.sin(re) * t_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 * (im_m ^ 3.0))
	tmp = 0.0
	if (im_m <= 330.0)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 5.5e+102)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * t_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 ^ 3.0);
	tmp = 0.0;
	if (im_m <= 330.0)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 5.5e+102)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = sin(re) * t_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[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 330.0], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.5e+102], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 330

   1. Initial program 49.7%

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

   3. Taylor expanded in im around 0 90.8%

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

      1. associate-*r* 90.8%

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

      2. neg-mul-1 90.8%

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

      3. associate-*r* 90.8%

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

      4. distribute-rgt-out 90.8%

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

      5. *-commutative 90.8%

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

   5. Simplified 90.8%

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

   6. Taylor expanded in re around inf 90.8%

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

   if 330 < im < 5.49999999999999981e102

   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 73.1%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

   5. Simplified 73.1%

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

   if 5.49999999999999981e102 < 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%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 330:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 6400:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 1.3 \cdot 10^{+101}:\\ \;\;\;\;im_m \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6400.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.3e+101)
      (* im_m (- (* 0.16666666666666666 (pow re 3.0)) re))
      (* (sin re) (* -0.16666666666666666 (pow im_m 3.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6400.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.3e+101) {
		tmp = im_m * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else {
		tmp = sin(re) * (-0.16666666666666666 * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6400.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.3d+101) then
        tmp = im_m * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else
        tmp = sin(re) * ((-0.16666666666666666d0) * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6400.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.3e+101) {
		tmp = im_m * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else {
		tmp = Math.sin(re) * (-0.16666666666666666 * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6400.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.3e+101:
		tmp = im_m * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	else:
		tmp = math.sin(re) * (-0.16666666666666666 * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6400.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.3e+101)
		tmp = Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	else
		tmp = Float64(sin(re) * Float64(-0.16666666666666666 * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6400.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.3e+101)
		tmp = im_m * ((0.16666666666666666 * (re ^ 3.0)) - re);
	else
		tmp = sin(re) * (-0.16666666666666666 * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6400.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.3e+101], N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6400

   1. Initial program 49.7%

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

   3. Taylor expanded in im around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if 6400 < im < 1.3e101

   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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 22.0%

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

      1. +-commutative 22.0%

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

      2. mul-1-neg 22.0%

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

      3. unsub-neg 22.0%

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

      4. *-commutative 22.0%

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

      5. associate-*r* 22.0%

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

      6. *-commutative 22.0%

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

      7. distribute-rgt-out-- 30.3%

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

   8. Simplified 30.3%

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

   if 1.3e101 < 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 95.9%

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

      1. associate-*r* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. associate-*r* 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. *-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in im around inf 95.9%

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

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

   8. Simplified 95.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6400:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+101}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 6400:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 3.8 \cdot 10^{+99}:\\ \;\;\;\;im_m \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6400.0)
    (* im_m (- (sin re)))
    (if (<= im_m 3.8e+99)
      (* im_m (- (* 0.16666666666666666 (pow re 3.0)) re))
      (* -0.16666666666666666 (* re (pow im_m 3.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6400.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 3.8e+99) {
		tmp = im_m * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6400.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 3.8d+99) then
        tmp = im_m * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6400.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 3.8e+99) {
		tmp = im_m * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6400.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 3.8e+99:
		tmp = im_m * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6400.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 3.8e+99)
		tmp = Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6400.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 3.8e+99)
		tmp = im_m * ((0.16666666666666666 * (re ^ 3.0)) - re);
	else
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6400.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 3.8e+99], N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6400

   1. Initial program 49.7%

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

   3. Taylor expanded in im around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if 6400 < im < 3.8e99

   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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 22.0%

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

      1. +-commutative 22.0%

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

      2. mul-1-neg 22.0%

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

      3. unsub-neg 22.0%

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

      4. *-commutative 22.0%

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

      5. associate-*r* 22.0%

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

      6. *-commutative 22.0%

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

      7. distribute-rgt-out-- 30.3%

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

   8. Simplified 30.3%

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

   if 3.8e99 < 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 95.9%

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

      1. associate-*r* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. associate-*r* 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. *-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in im around inf 95.9%

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

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

   8. Simplified 95.9%

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

   9. Taylor expanded in re around 0 75.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6400:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+99}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 6400:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 2.45 \cdot 10^{+97}:\\ \;\;\;\;im_m \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \end{array} \end{array} \]

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 6400.0)
    (* im_m (- (sin re)))
    (if (<= im_m 2.45e+97)
      (* im_m (- (* 0.16666666666666666 (pow re 3.0)) 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 <= 6400.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 2.45e+97) {
		tmp = im_m * ((0.16666666666666666 * pow(re, 3.0)) - 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 <= 6400.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 2.45d+97) then
        tmp = im_m * ((0.16666666666666666d0 * (re ** 3.0d0)) - 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 <= 6400.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 2.45e+97) {
		tmp = im_m * ((0.16666666666666666 * Math.pow(re, 3.0)) - 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 <= 6400.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 2.45e+97:
		tmp = im_m * ((0.16666666666666666 * math.pow(re, 3.0)) - 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 <= 6400.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 2.45e+97)
		tmp = Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - 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 <= 6400.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 2.45e+97)
		tmp = im_m * ((0.16666666666666666 * (re ^ 3.0)) - 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, 6400.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+97], N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $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 < 6400

   1. Initial program 49.7%

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

   3. Taylor expanded in im around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if 6400 < im < 2.44999999999999982e97

   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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 22.0%

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

      1. +-commutative 22.0%

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

      2. mul-1-neg 22.0%

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

      3. unsub-neg 22.0%

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

      4. *-commutative 22.0%

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

      5. associate-*r* 22.0%

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

      6. *-commutative 22.0%

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

      7. distribute-rgt-out-- 30.3%

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

   8. Simplified 30.3%

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

   if 2.44999999999999982e97 < 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 95.9%

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

      1. associate-*r* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. associate-*r* 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. *-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in re around 0 75.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6400:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+97}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 200000:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 1.76 \cdot 10^{+96}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im_m \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 200000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.76e+96)
      (* 0.16666666666666666 (* im_m (pow re 3.0)))
      (* -0.16666666666666666 (* re (pow im_m 3.0)))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 200000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.76e+96) {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 200000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.76d+96) then
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    else
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 200000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.76e+96) {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 200000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.76e+96:
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 200000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.76e+96)
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 200000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.76e+96)
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	else
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 200000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.76e+96], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2e5

   1. Initial program 50.0%

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

   3. Taylor expanded in im around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. neg-mul-1 70.8%

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

   5. Simplified 70.8%

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

   if 2e5 < im < 1.7599999999999999e96

   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 3.1%

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

      1. associate-*r* 3.1%

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

      2. neg-mul-1 3.1%

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

   5. Simplified 3.1%

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

   6. Taylor expanded in re around 0 22.8%

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

      1. +-commutative 22.8%

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

      2. mul-1-neg 22.8%

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

      3. unsub-neg 22.8%

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

      4. *-commutative 22.8%

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

      5. associate-*r* 22.8%

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

      6. *-commutative 22.8%

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

      7. distribute-rgt-out-- 31.5%

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

   8. Simplified 31.5%

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

   9. Taylor expanded in re around inf 31.0%

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

   if 1.7599999999999999e96 < 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 95.9%

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

      1. associate-*r* 95.9%

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

      2. neg-mul-1 95.9%

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

      3. associate-*r* 95.9%

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

      4. distribute-rgt-out 95.9%

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

      5. *-commutative 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in im around inf 95.9%

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

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

   8. Simplified 95.9%

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

   9. Taylor expanded in re around 0 75.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 200000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.76 \cdot 10^{+96}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.8e+20)
    (* im_m (- (sin re)))
    (* -0.16666666666666666 (* re (pow im_m 3.0))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8e+20) {
		tmp = im_m * -sin(re);
	} else {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.8d+20) then
        tmp = im_m * -sin(re)
    else
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8e+20) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.8e+20:
		tmp = im_m * -math.sin(re)
	else:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.8e+20)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.8e+20)
		tmp = im_m * -sin(re);
	else
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.8e+20], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.8e20

   1. Initial program 51.0%

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

   3. Taylor expanded in im around 0 69.4%

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

      1. associate-*r* 69.4%

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

      2. neg-mul-1 69.4%

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

   5. Simplified 69.4%

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

   if 4.8e20 < 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 68.7%

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

      1. associate-*r* 68.7%

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

      2. neg-mul-1 68.7%

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

      3. associate-*r* 68.7%

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

      4. distribute-rgt-out 68.7%

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

      5. *-commutative 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in im around inf 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in re around 0 57.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8 \cdot 10^{+20}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 1.85e+52) (* im_m (- (sin 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 (im_m <= 1.85e+52) {
		tmp = im_m * -sin(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 (im_m <= 1.85d+52) then
        tmp = im_m * -sin(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 (im_m <= 1.85e+52) {
		tmp = im_m * -Math.sin(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 im_m <= 1.85e+52:
		tmp = im_m * -math.sin(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 (im_m <= 1.85e+52)
		tmp = Float64(im_m * Float64(-sin(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 (im_m <= 1.85e+52)
		tmp = im_m * -sin(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[im$95$m, 1.85e+52], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.85e52

   1. Initial program 53.4%

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

   3. Taylor expanded in im around 0 66.1%

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

      1. associate-*r* 66.1%

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

      2. neg-mul-1 66.1%

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

   5. Simplified 66.1%

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

   if 1.85e52 < 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 4.7%

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

      1. associate-*r* 4.7%

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

      2. neg-mul-1 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in re around 0 15.8%

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

      1. associate-*r* 15.8%

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

      2. neg-mul-1 15.8%

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

   8. Simplified 15.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.8% 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 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 * 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 63.3%

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

3. Taylor expanded in im around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. neg-mul-1 53.2%

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

5. Simplified 53.2%

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

6. Taylor expanded in re around 0 29.4%

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

   1. associate-*r* 29.4%

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

   2. neg-mul-1 29.4%

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

8. Simplified 29.4%

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

9. Final simplification 29.4%

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

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot -3 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -3.0))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -3.0;
}

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 * (-3.0d0)
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 * -3.0;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -3.0

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -3.0)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -3.0;
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 * -3.0), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in im around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. neg-mul-1 53.2%

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

5. Simplified 53.2%

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

7. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-3} \]

8. Final simplification 2.9%

   \[\leadsto -3 \]
9. Add Preprocessing

Alternative 12: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot -0.004629629629629629 \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -0.004629629629629629))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -0.004629629629629629;
}

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 * (-0.004629629629629629d0)
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 * -0.004629629629629629;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -0.004629629629629629

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -0.004629629629629629)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -0.004629629629629629;
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 * -0.004629629629629629), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in im around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. neg-mul-1 53.2%

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

5. Simplified 53.2%

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

7. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-0.004629629629629629} \]

8. Final simplification 2.9%

   \[\leadsto -0.004629629629629629 \]
9. Add Preprocessing

Alternative 13: 15.1% 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 63.3%

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

3. Taylor expanded in im around 0 53.2%

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

   1. associate-*r* 53.2%

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

   2. neg-mul-1 53.2%

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

5. Simplified 53.2%

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

7. Applied egg-rr 15.1%

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

8. Final simplification 15.1%

   \[\leadsto 0 \]
9. 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 2024017 
(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))))