math.cos on complex, imaginary part

Percentage Accurate: 66.4% → 99.8%

Time: 8.1s

Alternatives: 11

Speedup: 2.8×

• 50.2% 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: 66.4% 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 57.1%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. neg-mul-1 87.0%

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

      3. associate-*r* 87.0%

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

      4. distribute-rgt-out 87.0%

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

      5. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in re around inf 87.0%

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

4. Final simplification 90.5%

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

Alternative 2: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im_m}^{3}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 0.42:\\ \;\;\;\;\sin re \cdot \left(t_0 - im_m\right)\\ \mathbf{elif}\;im_m \leq 6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im_m} - e^{im_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \end{array} \]

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 0.42)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 6e+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 <= 0.42) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 6e+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 <= 0.42d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 6d+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 <= 0.42) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 6e+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 <= 0.42:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 6e+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 <= 0.42)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 6e+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 <= 0.42)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 6e+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, 0.42], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6e+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 < 0.419999999999999984

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. neg-mul-1 87.0%

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

      3. associate-*r* 87.0%

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

      4. distribute-rgt-out 87.0%

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

      5. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in re around inf 87.0%

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

   if 0.419999999999999984 < im < 5.9999999999999996e102

   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 87.0%

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

      1. associate-*r* 87.0%

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

      2. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   if 5.9999999999999996e102 < 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 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.42:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 620:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im_m \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

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 620.0)
    (* im_m (- (sin re)))
    (if (<= im_m 5e+95)
      (* -0.16666666666666666 (* re (pow im_m 3.0)))
      (* (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 <= 620.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 5e+95) {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	} 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 <= 620.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 5d+95) then
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    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 <= 620.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 5e+95) {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	} 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 <= 620.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 5e+95:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	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 <= 620.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 5e+95)
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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 <= 620.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 5e+95)
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	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, 620.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 5e+95], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $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 < 620

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 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 620 < im < 5.00000000000000025e95

   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 90.9%

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

      1. associate-*r* 90.9%

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

      2. *-commutative 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in im around 0 20.8%

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

   7. Taylor expanded in im around inf 20.8%

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

   if 5.00000000000000025e95 < 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 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 620:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im_m}^{3}\\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 1850000:\\ \;\;\;\;\sin re \cdot \left(t_0 - im_m\right)\\ \mathbf{elif}\;im_m \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \end{array} \]

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 1850000.0)
      (* (sin re) (- t_0 im_m))
      (if (<= im_m 5e+95)
        (* -0.16666666666666666 (* re (pow im_m 3.0)))
        (* (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 <= 1850000.0) {
		tmp = sin(re) * (t_0 - im_m);
	} else if (im_m <= 5e+95) {
		tmp = -0.16666666666666666 * (re * pow(im_m, 3.0));
	} 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 <= 1850000.0d0) then
        tmp = sin(re) * (t_0 - im_m)
    else if (im_m <= 5d+95) then
        tmp = (-0.16666666666666666d0) * (re * (im_m ** 3.0d0))
    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 <= 1850000.0) {
		tmp = Math.sin(re) * (t_0 - im_m);
	} else if (im_m <= 5e+95) {
		tmp = -0.16666666666666666 * (re * Math.pow(im_m, 3.0));
	} 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 <= 1850000.0:
		tmp = math.sin(re) * (t_0 - im_m)
	elif im_m <= 5e+95:
		tmp = -0.16666666666666666 * (re * math.pow(im_m, 3.0))
	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 <= 1850000.0)
		tmp = Float64(sin(re) * Float64(t_0 - im_m));
	elseif (im_m <= 5e+95)
		tmp = Float64(-0.16666666666666666 * Float64(re * (im_m ^ 3.0)));
	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 <= 1850000.0)
		tmp = sin(re) * (t_0 - im_m);
	elseif (im_m <= 5e+95)
		tmp = -0.16666666666666666 * (re * (im_m ^ 3.0));
	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, 1850000.0], N[(N[Sin[re], $MachinePrecision] * N[(t$95$0 - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5e+95], N[(-0.16666666666666666 * N[(re * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 1.85e6

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. neg-mul-1 87.0%

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

      3. associate-*r* 87.0%

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

      4. distribute-rgt-out 87.0%

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

      5. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in re around inf 87.0%

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

   if 1.85e6 < im < 5.00000000000000025e95

   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 90.9%

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

      1. associate-*r* 90.9%

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

      2. *-commutative 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in im around 0 20.8%

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

   7. Taylor expanded in im around inf 20.8%

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

   if 5.00000000000000025e95 < 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 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1850000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 7000:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im_m}^{3} - im_m\right)\\ \end{array} \end{array} \]

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 7000.0)
    (* 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 <= 7000.0) {
		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 <= 7000.0d0) 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 <= 7000.0) {
		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 <= 7000.0:
		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 <= 7000.0)
		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 <= 7000.0)
		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, 7000.0], 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 < 7e3

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 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 7e3 < 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 68.1%

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

      1. associate-*r* 68.1%

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

      2. neg-mul-1 68.1%

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

      3. associate-*r* 68.1%

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

      4. distribute-rgt-out 68.1%

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

      5. *-commutative 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in re around 0 51.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 320:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im_m}^{3}\right)\\ \end{array} \end{array} \]

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 320.0)
    (* 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 <= 320.0) {
		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 <= 320.0d0) 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 <= 320.0) {
		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 <= 320.0:
		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 <= 320.0)
		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 <= 320.0)
		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, 320.0], 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 < 320

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 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 320 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 73.9%

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

      1. associate-*r* 73.9%

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

      2. *-commutative 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in im around 0 51.6%

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

   7. Taylor expanded in im around inf 51.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 320:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im_s \cdot \begin{array}{l} \mathbf{if}\;im_m \leq 390:\\ \;\;\;\;im_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

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 390.0) (* 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 <= 390.0) {
		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 <= 390.0d0) 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 <= 390.0) {
		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 <= 390.0:
		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 <= 390.0)
		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 <= 390.0)
		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, 390.0], 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 < 390

   1. Initial program 57.1%

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

   3. Taylor expanded in im around 0 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 390 < 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 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   5. Simplified 4.2%

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

   6. Taylor expanded in re around 0 11.4%

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

      1. associate-*r* 11.4%

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

      2. neg-mul-1 11.4%

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

   8. Simplified 11.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.4% 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 68.7%

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

3. Taylor expanded in im around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. neg-mul-1 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in re around 0 33.3%

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

   1. associate-*r* 33.3%

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

   2. neg-mul-1 33.3%

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

8. Simplified 33.3%

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

9. Final simplification 33.3%

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

Alternative 9: 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 68.7%

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

3. Taylor expanded in im around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. neg-mul-1 49.4%

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

5. Simplified 49.4%

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

7. Applied egg-rr 2.7%

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

8. Final simplification 2.7%

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

Alternative 10: 2.7% accurate, 308.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -0.16666666666666666))

C

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

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 * (-0.16666666666666666d0)
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 * -0.16666666666666666;
}

Python

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

Julia

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

MATLAB

im_m = abs(im);
im_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -0.16666666666666666;
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 * -0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in im around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. neg-mul-1 49.4%

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

5. Simplified 49.4%

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

7. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{-0.16666666666666666} \]

8. Final simplification 2.7%

   \[\leadsto -0.16666666666666666 \]
9. Add Preprocessing

Alternative 11: 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 68.7%

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

3. Taylor expanded in im around 0 49.4%

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

   1. associate-*r* 49.4%

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

   2. neg-mul-1 49.4%

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

5. Simplified 49.4%

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

7. Applied egg-rr 15.8%

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

8. Final simplification 15.8%

   \[\leadsto 0 \]
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 2024024 
(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))))