math.cos on complex, imaginary part

Percentage Accurate: 65.0% → 99.8%

Time: 11.0s

Alternatives: 12

Speedup: 2.7×

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

Specification

Math

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

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: 65.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.2)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.2d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.2:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.2], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. associate-*r* 93.6%

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

      4. distribute-rgt-out 93.6%

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

      5. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in re around inf 93.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im\_m}^{3}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.07:\\ \;\;\;\;\sin re \cdot \left(t\_0 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 0.07)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 2.45e+95)
        (* (- (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 <= 0.07) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 2.45e+95) {
		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 <= 0.07d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 2.45d+95) 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 <= 0.07) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 2.45e+95) {
		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 <= 0.07:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 2.45e+95:
		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 <= 0.07)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 2.45e+95)
		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 <= 0.07)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 2.45e+95)
		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, 0.07], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+95], 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 < 0.070000000000000007

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. associate-*r* 93.6%

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

      4. distribute-rgt-out 93.6%

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

      5. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in re around inf 93.6%

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

   if 0.070000000000000007 < im < 2.4499999999999999e95

   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 93.7%

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

      1. associate-*r* 93.7%

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

      2. *-commutative 93.7%

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

   5. Simplified 93.7%

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

   if 2.4499999999999999e95 < 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 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

      3. associate-*r* 98.1%

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

      4. distribute-rgt-out 98.1%

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

      5. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in im around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.07:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 1.5× speedup

Math

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

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 2.5)
    (* im_m (- (sin 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 <= 2.5) {
		tmp = im_m * -sin(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 <= 2.5d0) then
        tmp = im_m * -sin(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 <= 2.5) {
		tmp = im_m * -Math.sin(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 <= 2.5:
		tmp = im_m * -math.sin(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 <= 2.5)
		tmp = Float64(im_m * Float64(-sin(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 <= 2.5)
		tmp = im_m * -sin(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, 2.5], N[(im$95$m * (-N[Sin[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 2 regimes

2. if im < 2.5

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. neg-mul-1 73.4%

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

   5. Simplified 73.4%

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

   if 2.5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. neg-mul-1 76.0%

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

      3. associate-*r* 76.0%

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

      4. distribute-rgt-out 76.0%

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

      5. *-commutative 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in im around inf 76.0%

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

      1. associate-*r* 76.0%

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

      2. *-commutative 76.0%

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

   8. Simplified 76.0%

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

4. Final simplification 74.0%

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

Alternative 4: 83.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* (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) {
	return im_s * (sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m));
}

Fortran

im_m = abs(im)
im_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m))
end function

Java

im_m = Math.abs(im);
im_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m));
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m))

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m)))
end

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
im_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(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. Initial program 62.9%

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

3. Taylor expanded in im around 0 89.2%

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

   1. associate-*r* 89.2%

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

   2. neg-mul-1 89.2%

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

   3. associate-*r* 89.2%

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

   4. distribute-rgt-out 89.2%

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

   5. *-commutative 89.2%

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

5. Simplified 89.2%

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

6. Taylor expanded in re around inf 89.2%

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

7. Final simplification 89.2%

   \[\leadsto \sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]
8. Add Preprocessing

Alternative 5: 58.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+203} \lor \neg \left(im\_m \leq 7.4 \cdot 10^{+258}\right):\\ \;\;\;\;im\_m \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

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 2.2e+65)
    (* im_m (- (sin re)))
    (if (or (<= im_m 2.8e+203) (not (<= im_m 7.4e+258)))
      (* im_m (- (* 0.16666666666666666 (pow re 3.0)) 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 <= 2.2e+65) {
		tmp = im_m * -sin(re);
	} else if ((im_m <= 2.8e+203) || !(im_m <= 7.4e+258)) {
		tmp = im_m * ((0.16666666666666666 * pow(re, 3.0)) - 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 <= 2.2d+65) then
        tmp = im_m * -sin(re)
    else if ((im_m <= 2.8d+203) .or. (.not. (im_m <= 7.4d+258))) then
        tmp = im_m * ((0.16666666666666666d0 * (re ** 3.0d0)) - 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 <= 2.2e+65) {
		tmp = im_m * -Math.sin(re);
	} else if ((im_m <= 2.8e+203) || !(im_m <= 7.4e+258)) {
		tmp = im_m * ((0.16666666666666666 * Math.pow(re, 3.0)) - 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 <= 2.2e+65:
		tmp = im_m * -math.sin(re)
	elif (im_m <= 2.8e+203) or not (im_m <= 7.4e+258):
		tmp = im_m * ((0.16666666666666666 * math.pow(re, 3.0)) - 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 <= 2.2e+65)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif ((im_m <= 2.8e+203) || !(im_m <= 7.4e+258))
		tmp = Float64(im_m * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - 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 <= 2.2e+65)
		tmp = im_m * -sin(re);
	elseif ((im_m <= 2.8e+203) || ~((im_m <= 7.4e+258)))
		tmp = im_m * ((0.16666666666666666 * (re ^ 3.0)) - 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, 2.2e+65], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 2.8e+203], N[Not[LessEqual[im$95$m, 7.4e+258]], $MachinePrecision]], N[(im$95$m * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.1999999999999998e65

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   if 2.1999999999999998e65 < im < 2.7999999999999999e203 or 7.3999999999999996e258 < 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 5.3%

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

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around 0 9.5%

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

      1. +-commutative 9.5%

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

      2. mul-1-neg 9.5%

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

      3. unsub-neg 9.5%

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

      4. *-commutative 9.5%

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

      5. associate-*r* 9.5%

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

      6. *-commutative 9.5%

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

      7. distribute-rgt-out-- 35.2%

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

   8. Simplified 35.2%

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

   if 2.7999999999999999e203 < im < 7.3999999999999996e258

   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 5.3%

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

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around 0 36.0%

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

      1. associate-*r* 36.0%

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

      2. neg-mul-1 36.0%

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

   8. Simplified 36.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{+65}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+203} \lor \neg \left(im \leq 7.4 \cdot 10^{+258}\right):\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 4.8 \cdot 10^{+201} \lor \neg \left(im\_m \leq 7.4 \cdot 10^{+258}\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

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 2.9e+65)
    (* im_m (- (sin re)))
    (if (or (<= im_m 4.8e+201) (not (<= im_m 7.4e+258)))
      (* 0.16666666666666666 (* im_m (pow re 3.0)))
      (* 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 <= 2.9e+65) {
		tmp = im_m * -sin(re);
	} else if ((im_m <= 4.8e+201) || !(im_m <= 7.4e+258)) {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	} 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 <= 2.9d+65) then
        tmp = im_m * -sin(re)
    else if ((im_m <= 4.8d+201) .or. (.not. (im_m <= 7.4d+258))) then
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    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 <= 2.9e+65) {
		tmp = im_m * -Math.sin(re);
	} else if ((im_m <= 4.8e+201) || !(im_m <= 7.4e+258)) {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	} 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 <= 2.9e+65:
		tmp = im_m * -math.sin(re)
	elif (im_m <= 4.8e+201) or not (im_m <= 7.4e+258):
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	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 <= 2.9e+65)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif ((im_m <= 4.8e+201) || !(im_m <= 7.4e+258))
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	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 <= 2.9e+65)
		tmp = im_m * -sin(re);
	elseif ((im_m <= 4.8e+201) || ~((im_m <= 7.4e+258)))
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	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, 2.9e+65], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im$95$m, 4.8e+201], N[Not[LessEqual[im$95$m, 7.4e+258]], $MachinePrecision]], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.9e65

   1. Initial program 53.0%

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

   3. Taylor expanded in im around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   if 2.9e65 < im < 4.79999999999999985e201 or 7.3999999999999996e258 < 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 5.3%

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

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around 0 9.5%

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

      1. +-commutative 9.5%

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

      2. mul-1-neg 9.5%

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

      3. unsub-neg 9.5%

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

   8. Simplified 9.5%

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

   9. Taylor expanded in re around inf 34.1%

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

   if 4.79999999999999985e201 < im < 7.3999999999999996e258

   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 5.3%

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

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in re around 0 36.0%

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

      1. associate-*r* 36.0%

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

      2. neg-mul-1 36.0%

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

   8. Simplified 36.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.9 \cdot 10^{+65}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+201} \lor \neg \left(im \leq 7.4 \cdot 10^{+258}\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.165:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \]

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 0.165)
    (* im_m (- (sin 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 <= 0.165) {
		tmp = im_m * -sin(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 <= 0.165d0) then
        tmp = im_m * -sin(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 <= 0.165) {
		tmp = im_m * -Math.sin(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 <= 0.165:
		tmp = im_m * -math.sin(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 <= 0.165)
		tmp = Float64(im_m * Float64(-sin(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 <= 0.165)
		tmp = im_m * -sin(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, 0.165], N[(im$95$m * (-N[Sin[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 2 regimes

2. if im < 0.165000000000000008

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. neg-mul-1 73.4%

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

   5. Simplified 73.4%

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

   if 0.165000000000000008 < 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 76.0%

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

      1. associate-*r* 76.0%

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

      2. neg-mul-1 76.0%

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

      3. associate-*r* 76.0%

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

      4. distribute-rgt-out 76.0%

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

      5. *-commutative 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in re around 0 55.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.165:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 2.8× speedup

Math

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

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 0.165) (* 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 <= 0.165) {
		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 <= 0.165d0) 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 <= 0.165) {
		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 <= 0.165:
		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 <= 0.165)
		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 <= 0.165)
		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, 0.165], 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 < 0.165000000000000008

   1. Initial program 50.5%

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

   3. Taylor expanded in im around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. neg-mul-1 73.4%

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

   5. Simplified 73.4%

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

   if 0.165000000000000008 < 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 4.9%

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

      1. associate-*r* 4.9%

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

      2. neg-mul-1 4.9%

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

   5. Simplified 4.9%

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

   6. Taylor expanded in re around 0 16.5%

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

      1. associate-*r* 16.5%

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

      2. neg-mul-1 16.5%

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

   8. Simplified 16.5%

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

4. Final simplification 59.1%

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

Alternative 9: 32.1% 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 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 62.9%

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

3. Taylor expanded in im around 0 56.2%

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

   1. associate-*r* 56.2%

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

   2. neg-mul-1 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in re around 0 34.6%

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

   1. associate-*r* 34.6%

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

   2. neg-mul-1 34.6%

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

8. Simplified 34.6%

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

9. Final simplification 34.6%

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

Alternative 10: 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 62.9%

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

3. Taylor expanded in im around 0 56.2%

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

   1. associate-*r* 56.2%

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

   2. neg-mul-1 56.2%

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

5. Simplified 56.2%

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

7. Applied egg-rr 2.9%

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

8. Final simplification 2.9%

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

Alternative 11: 2.7% accurate, 308.0× speedup

Math

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

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 62.9%

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

3. Taylor expanded in im around 0 56.2%

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

   1. associate-*r* 56.2%

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

   2. neg-mul-1 56.2%

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

5. Simplified 56.2%

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

7. Applied egg-rr 2.9%

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

8. Final simplification 2.9%

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

Alternative 12: 14.2% 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 62.9%

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

3. Taylor expanded in im around 0 56.2%

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

   1. associate-*r* 56.2%

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

   2. neg-mul-1 56.2%

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

5. Simplified 56.2%

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

7. Applied egg-rr 16.3%

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

8. Final simplification 16.3%

   \[\leadsto 0 \]
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 2024026 
(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))))