math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.5%

Time: 15.3s

Alternatives: 15

Speedup: 2.7×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.2% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot \left(-0.008333333333333333 \cdot {im\_m}^{2} - 0.16666666666666666\right) - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -1e+26)
      (* t_0 (* 0.5 (sin re)))
      (*
       (sin re)
       (-
        (*
         (pow im_m 3.0)
         (- (* -0.008333333333333333 (pow im_m 2.0)) 0.16666666666666666))
        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 <= -1e+26) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * ((-0.008333333333333333 * pow(im_m, 2.0)) - 0.16666666666666666)) - 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 <= (-1d+26)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (((-0.008333333333333333d0) * (im_m ** 2.0d0)) - 0.16666666666666666d0)) - 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 <= -1e+26) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * ((-0.008333333333333333 * Math.pow(im_m, 2.0)) - 0.16666666666666666)) - 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 <= -1e+26:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * ((-0.008333333333333333 * math.pow(im_m, 2.0)) - 0.16666666666666666)) - 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 <= -1e+26)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * Float64(Float64(-0.008333333333333333 * (im_m ^ 2.0)) - 0.16666666666666666)) - 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 <= -1e+26)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * ((-0.008333333333333333 * (im_m ^ 2.0)) - 0.16666666666666666)) - 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, -1e+26], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[(-0.008333333333333333 * N[Power[im$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $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)) < -1.00000000000000005e26

   1. Initial program 100.0%

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

   if -1.00000000000000005e26 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 89.8%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 89.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re\right) \cdot im + \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. +-commutative 89.8%

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im + \left(-1 \cdot \sin re\right) \cdot im} \]

      3. *-commutative 89.8%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} + \left(-1 \cdot \sin re\right) \cdot im \]

      4. mul-1-neg 89.8%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(-\sin re\right)} \cdot im \]

      5. cancel-sign-sub-inv 89.8%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - \sin re \cdot im} \]

      6. associate-*r* 89.8%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - \sin re \cdot im \]

      7. *-commutative 89.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - \sin re \cdot im \]

      8. associate-*r* 89.8%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      9. distribute-rgt-out 89.8%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      10. associate-*l* 90.8%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - \sin re \cdot im \]

      11. distribute-lft-out-- 90.8%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right) - im\right)} \]

   5. Simplified 90.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in re around inf 90.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot \left(-0.008333333333333333 \cdot {im}^{2} - 0.16666666666666666\right) - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -1e+26)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) 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 <= -1e+26) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - 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 <= (-1d+26)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - 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 <= -1e+26) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - 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 <= -1e+26:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - 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 <= -1e+26)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - 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 <= -1e+26)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - 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, -1e+26], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $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)) < -1.00000000000000005e26

   1. Initial program 100.0%

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

   if -1.00000000000000005e26 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. mul-1-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-*r* 84.7%

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

      6. distribute-lft-out-- 84.7%

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

      7. associate-*r* 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.7%

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

      10. associate-*r* 85.2%

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

      11. distribute-rgt-out-- 85.2%

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

      12. unsub-neg 85.2%

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

      13. unsub-neg 85.2%

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

   5. Simplified 85.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 3.5 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;re \cdot \left(im\_m + -0.16666666666666666 \cdot \left(im\_m \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (log1p (expm1 re))))
   (*
    im_s
    (if (<= im_m 10000000.0)
      (* im_m (- (sin re)))
      (if (<= im_m 3.5e+24)
        t_0
        (if (<= im_m 1.3e+44)
          (* re (+ im_m (* -0.16666666666666666 (* im_m (pow re 2.0)))))
          (if (<= im_m 4.5e+51)
            t_0
            (* -0.008333333333333333 (* (sin re) (pow im_m 5.0))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = log1p(expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 3.5e+24) {
		tmp = t_0;
	} else if (im_m <= 1.3e+44) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * pow(re, 2.0))));
	} else if (im_m <= 4.5e+51) {
		tmp = t_0;
	} else {
		tmp = -0.008333333333333333 * (sin(re) * pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.log1p(Math.expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 3.5e+24) {
		tmp = t_0;
	} else if (im_m <= 1.3e+44) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * Math.pow(re, 2.0))));
	} else if (im_m <= 4.5e+51) {
		tmp = t_0;
	} else {
		tmp = -0.008333333333333333 * (Math.sin(re) * Math.pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.log1p(math.expm1(re))
	tmp = 0
	if im_m <= 10000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 3.5e+24:
		tmp = t_0
	elif im_m <= 1.3e+44:
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * math.pow(re, 2.0))))
	elif im_m <= 4.5e+51:
		tmp = t_0
	else:
		tmp = -0.008333333333333333 * (math.sin(re) * math.pow(im_m, 5.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = log1p(expm1(re))
	tmp = 0.0
	if (im_m <= 10000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 3.5e+24)
		tmp = t_0;
	elseif (im_m <= 1.3e+44)
		tmp = Float64(re * Float64(im_m + Float64(-0.16666666666666666 * Float64(im_m * (re ^ 2.0)))));
	elseif (im_m <= 4.5e+51)
		tmp = t_0;
	else
		tmp = Float64(-0.008333333333333333 * Float64(sin(re) * (im_m ^ 5.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 10000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 3.5e+24], t$95$0, If[LessEqual[im$95$m, 1.3e+44], N[(re * N[(im$95$m + N[(-0.16666666666666666 * N[(im$95$m * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+51], t$95$0, N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if im < 1e7

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 1e7 < im < 3.5000000000000002e24 or 1.3e44 < im < 4.5e51

   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 20.0%

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

      1. associate-*r* 20.0%

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

      2. *-commutative 20.0%

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

   5. Simplified 20.0%

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

   7. Applied egg-rr 80.2%

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

   if 3.5000000000000002e24 < im < 1.3e44

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   5. Simplified 3.4%

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

      1. pow1 3.4%

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

      2. *-commutative 3.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.6%

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

      5. sqr-neg 0.6%

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

      6. sqrt-prod 0.6%

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

      7. add-sqr-sqrt 0.6%

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

   7. Applied egg-rr 0.6%

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

      1. unpow1 0.6%

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

      2. *-commutative 0.6%

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

   9. Simplified 0.6%

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

   10. Taylor expanded in re around 0 55.0%

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

   if 4.5e51 < 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 90.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 90.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re\right) \cdot im + \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. +-commutative 90.3%

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im + \left(-1 \cdot \sin re\right) \cdot im} \]

      3. *-commutative 90.3%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} + \left(-1 \cdot \sin re\right) \cdot im \]

      4. mul-1-neg 90.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(-\sin re\right)} \cdot im \]

      5. cancel-sign-sub-inv 90.3%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - \sin re \cdot im} \]

      6. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - \sin re \cdot im \]

      7. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - \sin re \cdot im \]

      8. associate-*r* 91.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      9. distribute-rgt-out 91.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      10. associate-*l* 95.1%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - \sin re \cdot im \]

      11. distribute-lft-out-- 95.1%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right) - im\right)} \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 95.1%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;re \cdot \left(im + -0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10000000:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 6.1 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 2.4 \cdot 10^{+44}:\\ \;\;\;\;re \cdot \left(im\_m + -0.16666666666666666 \cdot \left(im\_m \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (log1p (expm1 re))))
   (*
    im_s
    (if (<= im_m 10000000.0)
      (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
      (if (<= im_m 6.1e+28)
        t_0
        (if (<= im_m 2.4e+44)
          (* re (+ im_m (* -0.16666666666666666 (* im_m (pow re 2.0)))))
          (if (<= im_m 4.5e+51)
            t_0
            (* -0.008333333333333333 (* (sin re) (pow im_m 5.0))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = log1p(expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6.1e+28) {
		tmp = t_0;
	} else if (im_m <= 2.4e+44) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * pow(re, 2.0))));
	} else if (im_m <= 4.5e+51) {
		tmp = t_0;
	} else {
		tmp = -0.008333333333333333 * (sin(re) * pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.log1p(Math.expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 6.1e+28) {
		tmp = t_0;
	} else if (im_m <= 2.4e+44) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * Math.pow(re, 2.0))));
	} else if (im_m <= 4.5e+51) {
		tmp = t_0;
	} else {
		tmp = -0.008333333333333333 * (Math.sin(re) * Math.pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.log1p(math.expm1(re))
	tmp = 0
	if im_m <= 10000000.0:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 6.1e+28:
		tmp = t_0
	elif im_m <= 2.4e+44:
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * math.pow(re, 2.0))))
	elif im_m <= 4.5e+51:
		tmp = t_0
	else:
		tmp = -0.008333333333333333 * (math.sin(re) * math.pow(im_m, 5.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = log1p(expm1(re))
	tmp = 0.0
	if (im_m <= 10000000.0)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 6.1e+28)
		tmp = t_0;
	elseif (im_m <= 2.4e+44)
		tmp = Float64(re * Float64(im_m + Float64(-0.16666666666666666 * Float64(im_m * (re ^ 2.0)))));
	elseif (im_m <= 4.5e+51)
		tmp = t_0;
	else
		tmp = Float64(-0.008333333333333333 * Float64(sin(re) * (im_m ^ 5.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 10000000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 6.1e+28], t$95$0, If[LessEqual[im$95$m, 2.4e+44], N[(re * N[(im$95$m + N[(-0.16666666666666666 * N[(im$95$m * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+51], t$95$0, N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if im < 1e7

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. mul-1-neg 84.3%

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

      3. unsub-neg 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-*r* 84.3%

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

      6. distribute-lft-out-- 84.3%

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

      7. associate-*r* 84.3%

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

      8. *-commutative 84.3%

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

      9. associate-*r* 84.3%

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

      10. associate-*r* 84.8%

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

      11. distribute-rgt-out-- 84.8%

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

      12. unsub-neg 84.8%

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

      13. unsub-neg 84.8%

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

   5. Simplified 84.8%

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

   if 1e7 < im < 6.1000000000000002e28 or 2.40000000000000013e44 < im < 4.5e51

   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 20.0%

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

      1. associate-*r* 20.0%

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

      2. *-commutative 20.0%

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

   5. Simplified 20.0%

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

   7. Applied egg-rr 80.2%

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

   if 6.1000000000000002e28 < im < 2.40000000000000013e44

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   5. Simplified 3.4%

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

      1. pow1 3.4%

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

      2. *-commutative 3.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.6%

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

      5. sqr-neg 0.6%

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

      6. sqrt-prod 0.6%

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

      7. add-sqr-sqrt 0.6%

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

   7. Applied egg-rr 0.6%

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

      1. unpow1 0.6%

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

      2. *-commutative 0.6%

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

   9. Simplified 0.6%

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

   10. Taylor expanded in re around 0 55.0%

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

   if 4.5e51 < 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 90.3%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 90.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re\right) \cdot im + \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. +-commutative 90.3%

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im + \left(-1 \cdot \sin re\right) \cdot im} \]

      3. *-commutative 90.3%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} + \left(-1 \cdot \sin re\right) \cdot im \]

      4. mul-1-neg 90.3%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(-\sin re\right)} \cdot im \]

      5. cancel-sign-sub-inv 90.3%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - \sin re \cdot im} \]

      6. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - \sin re \cdot im \]

      7. *-commutative 91.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - \sin re \cdot im \]

      8. associate-*r* 91.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      9. distribute-rgt-out 91.9%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      10. associate-*l* 95.1%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - \sin re \cdot im \]

      11. distribute-lft-out-- 95.1%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right) - im\right)} \]

   5. Simplified 95.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 95.1%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10000000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 6.1 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+44}:\\ \;\;\;\;re \cdot \left(im + -0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;re \cdot \left(im\_m + -0.16666666666666666 \cdot \left(im\_m \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.02 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (log1p (expm1 re))))
   (*
    im_s
    (if (<= im_m 10000000.0)
      (* im_m (- (sin re)))
      (if (<= im_m 6.5e+25)
        t_0
        (if (<= im_m 2.75e+45)
          (* re (+ im_m (* -0.16666666666666666 (* im_m (pow re 2.0)))))
          (if (<= im_m 1.02e+90)
            t_0
            (* re (- (* (pow im_m 3.0) -0.16666666666666666) 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 = log1p(expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 6.5e+25) {
		tmp = t_0;
	} else if (im_m <= 2.75e+45) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * pow(re, 2.0))));
	} else if (im_m <= 1.02e+90) {
		tmp = t_0;
	} else {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.log1p(Math.expm1(re));
	double tmp;
	if (im_m <= 10000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 6.5e+25) {
		tmp = t_0;
	} else if (im_m <= 2.75e+45) {
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * Math.pow(re, 2.0))));
	} else if (im_m <= 1.02e+90) {
		tmp = t_0;
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - 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.log1p(math.expm1(re))
	tmp = 0
	if im_m <= 10000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 6.5e+25:
		tmp = t_0
	elif im_m <= 2.75e+45:
		tmp = re * (im_m + (-0.16666666666666666 * (im_m * math.pow(re, 2.0))))
	elif im_m <= 1.02e+90:
		tmp = t_0
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - 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 = log1p(expm1(re))
	tmp = 0.0
	if (im_m <= 10000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 6.5e+25)
		tmp = t_0;
	elseif (im_m <= 2.75e+45)
		tmp = Float64(re * Float64(im_m + Float64(-0.16666666666666666 * Float64(im_m * (re ^ 2.0)))));
	elseif (im_m <= 1.02e+90)
		tmp = t_0;
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 10000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 6.5e+25], t$95$0, If[LessEqual[im$95$m, 2.75e+45], N[(re * N[(im$95$m + N[(-0.16666666666666666 * N[(im$95$m * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.02e+90], t$95$0, N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if im < 1e7

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 1e7 < im < 6.50000000000000005e25 or 2.75e45 < im < 1.02000000000000005e90

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0 57.1%

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

      1. associate-*r* 57.1%

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

      2. *-commutative 57.1%

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

   5. Simplified 57.1%

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

   7. Applied egg-rr 43.3%

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

   if 6.50000000000000005e25 < im < 2.75e45

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   5. Simplified 3.4%

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

      1. pow1 3.4%

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

      2. *-commutative 3.4%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.6%

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

      5. sqr-neg 0.6%

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

      6. sqrt-prod 0.6%

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

      7. add-sqr-sqrt 0.6%

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

   7. Applied egg-rr 0.6%

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

      1. unpow1 0.6%

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

      2. *-commutative 0.6%

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

   9. Simplified 0.6%

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

   10. Taylor expanded in re around 0 55.0%

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

   if 1.02000000000000005e90 < 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 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. unsub-neg 88.6%

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

      4. *-commutative 88.6%

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

      5. associate-*r* 88.6%

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

      6. distribute-lft-out-- 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-*r* 88.6%

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

      10. associate-*r* 98.1%

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

      11. distribute-rgt-out-- 98.1%

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

      12. unsub-neg 98.1%

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

      13. unsub-neg 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in re around 0 71.4%

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

      1. *-commutative 71.4%

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

   8. Simplified 71.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;re \cdot \left(im + -0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.051:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im\_m}^{5}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.051)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 4.5e+61)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* -0.008333333333333333 (* (sin re) (pow im_m 5.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.051) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.008333333333333333 * (sin(re) * pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.051d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 4.5d+61) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (-0.008333333333333333d0) * (sin(re) * (im_m ** 5.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.051) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 4.5e+61) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.008333333333333333 * (Math.sin(re) * Math.pow(im_m, 5.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.051:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 4.5e+61:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = -0.008333333333333333 * (math.sin(re) * math.pow(im_m, 5.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.051)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 4.5e+61)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(-0.008333333333333333 * Float64(sin(re) * (im_m ^ 5.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.051)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 4.5e+61)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = -0.008333333333333333 * (sin(re) * (im_m ^ 5.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.051], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.5e+61], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0509999999999999967

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. mul-1-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. *-commutative 84.7%

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

      5. associate-*r* 84.7%

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

      6. distribute-lft-out-- 84.7%

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

      7. associate-*r* 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*r* 84.7%

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

      10. associate-*r* 85.2%

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

      11. distribute-rgt-out-- 85.2%

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

      12. unsub-neg 85.2%

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

      13. unsub-neg 85.2%

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

   5. Simplified 85.2%

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

   if 0.0509999999999999967 < im < 4.5e61

   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 63.6%

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

      1. associate-*r* 63.6%

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

      2. *-commutative 63.6%

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

   5. Simplified 63.6%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 95.0%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 95.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re\right) \cdot im + \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. +-commutative 95.0%

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im + \left(-1 \cdot \sin re\right) \cdot im} \]

      3. *-commutative 95.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} + \left(-1 \cdot \sin re\right) \cdot im \]

      4. mul-1-neg 95.0%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(-\sin re\right)} \cdot im \]

      5. cancel-sign-sub-inv 95.0%

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - \sin re \cdot im} \]

      6. associate-*r* 96.6%

         \[\leadsto \color{blue}{\left(im \cdot {im}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right)} - \sin re \cdot im \]

      7. *-commutative 96.6%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{2} \cdot \sin re\right)\right) \cdot \left(im \cdot {im}^{2}\right)} - \sin re \cdot im \]

      8. associate-*r* 96.6%

         \[\leadsto \left(-0.16666666666666666 \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{2}\right) \cdot \sin re}\right) \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      9. distribute-rgt-out 96.6%

         \[\leadsto \color{blue}{\left(\sin re \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right)\right)} \cdot \left(im \cdot {im}^{2}\right) - \sin re \cdot im \]

      10. associate-*l* 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right)\right)} - \sin re \cdot im \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 + -0.008333333333333333 \cdot {im}^{2}\right) \cdot \left(im \cdot {im}^{2}\right) - im\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\mathsf{fma}\left({im}^{2}, -0.008333333333333333, -0.16666666666666666\right) \cdot {im}^{3} - im\right)} \]

   6. Taylor expanded in im around inf 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.051:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 13200000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 3.8 \cdot 10^{+98}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ \mathbf{elif}\;im\_m \leq 1.12 \cdot 10^{+282}:\\ \;\;\;\;im\_m \cdot \frac{-2.4}{\sin re}\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 13200000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 3.8e+98)
      (* 0.16666666666666666 (* im_m (pow re 3.0)))
      (if (<= im_m 1.12e+282) (* im_m (/ -2.4 (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 <= 13200000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 3.8e+98) {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	} else if (im_m <= 1.12e+282) {
		tmp = im_m * (-2.4 / 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 <= 13200000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 3.8d+98) then
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    else if (im_m <= 1.12d+282) then
        tmp = im_m * ((-2.4d0) / 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 <= 13200000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 3.8e+98) {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	} else if (im_m <= 1.12e+282) {
		tmp = im_m * (-2.4 / 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 <= 13200000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 3.8e+98:
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	elif im_m <= 1.12e+282:
		tmp = im_m * (-2.4 / 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 <= 13200000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 3.8e+98)
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	elseif (im_m <= 1.12e+282)
		tmp = Float64(im_m * Float64(-2.4 / 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 <= 13200000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 3.8e+98)
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	elseif (im_m <= 1.12e+282)
		tmp = im_m * (-2.4 / 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, 13200000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 3.8e+98], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.12e+282], N[(im$95$m * N[(-2.4 / N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 1.32e7

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 1.32e7 < im < 3.7999999999999999e98

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 3.2%

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

      1. associate-*r* 3.2%

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

      2. neg-mul-1 3.2%

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

   5. Simplified 3.2%

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

   6. Taylor expanded in re around 0 19.5%

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

      1. mul-1-neg 19.5%

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

      2. +-commutative 19.5%

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

      3. unsub-neg 19.5%

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

      4. *-commutative 19.5%

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

      5. *-commutative 19.5%

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

      6. associate-*r* 19.5%

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

   8. Simplified 19.5%

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

   9. Taylor expanded in re around inf 19.0%

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

   if 3.7999999999999999e98 < im < 1.12e282

   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 88.0%

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

      1. associate-*r* 88.0%

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

      2. distribute-rgt-out 88.0%

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

      3. *-commutative 88.0%

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

   5. Simplified 88.0%

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

      1. log1p-expm1-u 95.1%

         \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

      2. +-commutative 95.1%

         \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

      3. fma-define 95.1%

         \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

   7. Applied egg-rr 95.1%

      \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 95.1%

         \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)}}\right) \]

      2. pow3 95.1%

         \[\leadsto im \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

   9. Applied egg-rr 95.1%

      \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

   11. Applied egg-rr 32.2%

       \[\leadsto im \cdot \color{blue}{\frac{-2}{0.8333333333333334 \cdot \sin re}} \]
   12. Step-by-step derivation

       1. associate-/r* 32.2%

          \[\leadsto im \cdot \color{blue}{\frac{\frac{-2}{0.8333333333333334}}{\sin re}} \]

       2. metadata-eval 32.2%

          \[\leadsto im \cdot \frac{\color{blue}{-2.4}}{\sin re} \]

   13. Simplified 32.2%

       \[\leadsto im \cdot \color{blue}{\frac{-2.4}{\sin re}} \]

   if 1.12e282 < 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 10.4%

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

      1. associate-*r* 10.4%

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

      2. neg-mul-1 10.4%

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

   5. Simplified 10.4%

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

   6. Taylor expanded in re around 0 68.0%

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

      1. associate-*r* 68.0%

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

      2. mul-1-neg 68.0%

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

   8. Simplified 68.0%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 13200000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+98}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{elif}\;im \leq 1.12 \cdot 10^{+282}:\\ \;\;\;\;im \cdot \frac{-2.4}{\sin re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 280:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 9.2 \cdot 10^{+281}:\\ \;\;\;\;im\_m \cdot \frac{-2.4}{\sin re}\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 280.0)
    (* im_m (- (sin re)))
    (if (<= im_m 9.2e+281) (* im_m (/ -2.4 (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 <= 280.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 9.2e+281) {
		tmp = im_m * (-2.4 / 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 <= 280.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 9.2d+281) then
        tmp = im_m * ((-2.4d0) / 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 <= 280.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 9.2e+281) {
		tmp = im_m * (-2.4 / 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 <= 280.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 9.2e+281:
		tmp = im_m * (-2.4 / 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 <= 280.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 9.2e+281)
		tmp = Float64(im_m * Float64(-2.4 / 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 <= 280.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 9.2e+281)
		tmp = im_m * (-2.4 / 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, 280.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 9.2e+281], N[(im$95$m * N[(-2.4 / N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 280

   1. Initial program 58.1%

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

   3. Taylor expanded in im around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. neg-mul-1 64.5%

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

   5. Simplified 64.5%

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

   if 280 < im < 9.2000000000000003e281

   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 62.7%

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

      1. associate-*r* 62.7%

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

      2. distribute-rgt-out 62.7%

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

      3. *-commutative 62.7%

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

   5. Simplified 62.7%

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

      1. log1p-expm1-u 91.4%

         \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

      2. +-commutative 91.4%

         \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

      3. fma-define 91.4%

         \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

   7. Applied egg-rr 91.4%

      \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 91.3%

         \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)}}\right) \]

      2. pow3 91.3%

         \[\leadsto im \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

   9. Applied egg-rr 91.3%

      \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

   11. Applied egg-rr 23.8%

       \[\leadsto im \cdot \color{blue}{\frac{-2}{0.8333333333333334 \cdot \sin re}} \]
   12. Step-by-step derivation

       1. associate-/r* 23.8%

          \[\leadsto im \cdot \color{blue}{\frac{\frac{-2}{0.8333333333333334}}{\sin re}} \]

       2. metadata-eval 23.8%

          \[\leadsto im \cdot \frac{\color{blue}{-2.4}}{\sin re} \]

   13. Simplified 23.8%

       \[\leadsto im \cdot \color{blue}{\frac{-2.4}{\sin re}} \]

   if 9.2000000000000003e281 < 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 10.4%

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

      1. associate-*r* 10.4%

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

      2. neg-mul-1 10.4%

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

   5. Simplified 10.4%

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

   6. Taylor expanded in re around 0 68.0%

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

      1. associate-*r* 68.0%

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

      2. mul-1-neg 68.0%

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

   8. Simplified 68.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 280:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.2 \cdot 10^{+281}:\\ \;\;\;\;im \cdot \frac{-2.4}{\sin re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00068:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00068)
    (* im_m (- (sin re)))
    (* re (- (* (pow im_m 3.0) -0.16666666666666666) 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 <= 0.00068) {
		tmp = im_m * -sin(re);
	} else {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - 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 <= 0.00068d0) then
        tmp = im_m * -sin(re)
    else
        tmp = re * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - 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 <= 0.00068) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - 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 <= 0.00068:
		tmp = im_m * -math.sin(re)
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - 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 <= 0.00068)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - 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 <= 0.00068)
		tmp = im_m * -sin(re);
	else
		tmp = re * (((im_m ^ 3.0) * -0.16666666666666666) - 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, 0.00068], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.8e-4

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   5. Simplified 64.8%

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

   if 6.8e-4 < 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 67.0%

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

      1. +-commutative 67.0%

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

      2. mul-1-neg 67.0%

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

      3. unsub-neg 67.0%

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

      4. *-commutative 67.0%

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

      5. associate-*r* 67.0%

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

      6. distribute-lft-out-- 67.0%

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

      7. associate-*r* 67.0%

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

      8. *-commutative 67.0%

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

      9. associate-*r* 67.0%

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

      10. associate-*r* 74.0%

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

      11. distribute-rgt-out-- 74.0%

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

      12. unsub-neg 74.0%

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

      13. unsub-neg 74.0%

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

   5. Simplified 74.0%

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

   6. Taylor expanded in re around 0 56.7%

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

      1. *-commutative 56.7%

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

   8. Simplified 56.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00068:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.001:\\ \;\;\;\;im\_m \cdot re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= (sin re) -0.001) (* im_m re) (* im_m (- re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.001) {
		tmp = im_m * re;
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (sin(re) <= (-0.001d0)) then
        tmp = im_m * re
    else
        tmp = im_m * -re
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (Math.sin(re) <= -0.001) {
		tmp = im_m * re;
	} else {
		tmp = im_m * -re;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if math.sin(re) <= -0.001:
		tmp = im_m * re
	else:
		tmp = im_m * -re
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.001)
		tmp = Float64(im_m * re);
	else
		tmp = Float64(im_m * Float64(-re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (sin(re) <= -0.001)
		tmp = im_m * re;
	else
		tmp = im_m * -re;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[Sin[re], $MachinePrecision], -0.001], N[(im$95$m * re), $MachinePrecision], N[(im$95$m * (-re)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -1e-3

   1. Initial program 63.1%

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

   3. Taylor expanded in im around 0 44.4%

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

      1. associate-*r* 44.4%

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

      2. neg-mul-1 44.4%

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

   5. Simplified 44.4%

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

      1. pow1 44.4%

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

      2. *-commutative 44.4%

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

      3. add-sqr-sqrt 21.1%

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

      4. sqrt-unprod 23.7%

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

      5. sqr-neg 23.7%

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

      6. sqrt-prod 0.9%

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

      7. add-sqr-sqrt 1.6%

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

   7. Applied egg-rr 1.6%

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

      1. unpow1 1.6%

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

      2. *-commutative 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in re around 0 10.6%

       \[\leadsto \color{blue}{im \cdot re} \]
   11. Step-by-step derivation

       1. *-commutative 10.6%

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

   12. Simplified 10.6%

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

   if -1e-3 < (sin.f64 re)

   1. Initial program 70.6%

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

   3. Taylor expanded in im around 0 51.0%

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

      1. associate-*r* 51.0%

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

      2. neg-mul-1 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in re around 0 38.3%

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

      1. associate-*r* 38.3%

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

      2. mul-1-neg 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.001:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 60:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 60.0) (* 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 <= 60.0) {
		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 <= 60.0d0) 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 <= 60.0) {
		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 <= 60.0:
		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 <= 60.0)
		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 <= 60.0)
		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, 60.0], 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 < 60

   1. Initial program 57.9%

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

   3. Taylor expanded in im around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   5. Simplified 64.8%

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

   if 60 < 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 5.0%

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

      1. associate-*r* 5.0%

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

      2. neg-mul-1 5.0%

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

   5. Simplified 5.0%

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

   6. Taylor expanded in re around 0 19.0%

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

      1. associate-*r* 19.0%

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

      2. mul-1-neg 19.0%

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

   8. Simplified 19.0%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 5.5% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -2.5\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -2.5)))

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 * -2.5);
}

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 * (-2.5d0))
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 * -2.5);
}

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 * -2.5)

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 * -2.5))
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 * -2.5);
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 * -2.5), $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 80.1%

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

   1. associate-*r* 80.1%

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

   2. distribute-rgt-out 80.1%

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

   3. *-commutative 80.1%

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

5. Simplified 80.1%

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

   1. log1p-expm1-u 94.2%

      \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

   2. +-commutative 94.2%

      \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

   3. fma-define 94.2%

      \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

7. Applied egg-rr 94.2%

   \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]
8. Step-by-step derivation

   1. add-cbrt-cube 89.8%

      \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)}}\right) \]

   2. pow3 89.8%

      \[\leadsto im \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

9. Applied egg-rr 89.8%

   \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

11. Applied egg-rr 5.9%

    \[\leadsto im \cdot \color{blue}{\left(\left(0.8333333333333334 \cdot \sin re\right) \cdot \frac{-2}{-0.16666666666666666 \cdot \sin re + 0.8333333333333334 \cdot \sin re}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 5.9%

       \[\leadsto im \cdot \color{blue}{\frac{\left(0.8333333333333334 \cdot \sin re\right) \cdot -2}{-0.16666666666666666 \cdot \sin re + 0.8333333333333334 \cdot \sin re}} \]

    2. *-commutative 5.9%

       \[\leadsto im \cdot \frac{\color{blue}{-2 \cdot \left(0.8333333333333334 \cdot \sin re\right)}}{-0.16666666666666666 \cdot \sin re + 0.8333333333333334 \cdot \sin re} \]

    3. associate-/l* 5.9%

       \[\leadsto im \cdot \color{blue}{\left(-2 \cdot \frac{0.8333333333333334 \cdot \sin re}{-0.16666666666666666 \cdot \sin re + 0.8333333333333334 \cdot \sin re}\right)} \]

    4. *-commutative 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \frac{\color{blue}{\sin re \cdot 0.8333333333333334}}{-0.16666666666666666 \cdot \sin re + 0.8333333333333334 \cdot \sin re}\right) \]

    5. distribute-rgt-out 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \frac{\sin re \cdot 0.8333333333333334}{\color{blue}{\sin re \cdot \left(-0.16666666666666666 + 0.8333333333333334\right)}}\right) \]

    6. times-frac 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \color{blue}{\left(\frac{\sin re}{\sin re} \cdot \frac{0.8333333333333334}{-0.16666666666666666 + 0.8333333333333334}\right)}\right) \]

    7. *-inverses 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \left(\color{blue}{1} \cdot \frac{0.8333333333333334}{-0.16666666666666666 + 0.8333333333333334}\right)\right) \]

    8. metadata-eval 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \left(1 \cdot \frac{0.8333333333333334}{\color{blue}{0.6666666666666666}}\right)\right) \]

    9. metadata-eval 5.9%

       \[\leadsto im \cdot \left(-2 \cdot \left(1 \cdot \color{blue}{1.25}\right)\right) \]

    10. metadata-eval 5.9%

        \[\leadsto im \cdot \left(-2 \cdot \color{blue}{1.25}\right) \]

    11. metadata-eval 5.9%

        \[\leadsto im \cdot \color{blue}{-2.5} \]

13. Simplified 5.9%

    \[\leadsto im \cdot \color{blue}{-2.5} \]

14. Final simplification 5.9%

    \[\leadsto im \cdot -2.5 \]
15. Add Preprocessing

Alternative 13: 5.7% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot -1.4\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m -1.4)))

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 * -1.4);
}

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 * (-1.4d0))
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 * -1.4);
}

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 * -1.4)

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 * -1.4))
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 * -1.4);
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 * -1.4), $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 80.1%

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

   1. associate-*r* 80.1%

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

   2. distribute-rgt-out 80.1%

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

   3. *-commutative 80.1%

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

5. Simplified 80.1%

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

   1. log1p-expm1-u 94.2%

      \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

   2. +-commutative 94.2%

      \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

   3. fma-define 94.2%

      \[\leadsto im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

7. Applied egg-rr 94.2%

   \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]
8. Step-by-step derivation

   1. add-cbrt-cube 89.8%

      \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \cdot \mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)}}\right) \]

   2. pow3 89.8%

      \[\leadsto im \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

9. Applied egg-rr 89.8%

   \[\leadsto im \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\mathsf{expm1}\left(\sin re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)}^{3}}}\right) \]

11. Applied egg-rr 6.1%

    \[\leadsto im \cdot \color{blue}{\left(\frac{-0.16666666666666666 \cdot \sin re}{0.8333333333333334 \cdot \sin re} - \frac{\sin re}{0.8333333333333334 \cdot \sin re}\right)} \]
12. Step-by-step derivation

    1. times-frac 6.1%

       \[\leadsto im \cdot \left(\color{blue}{\frac{-0.16666666666666666}{0.8333333333333334} \cdot \frac{\sin re}{\sin re}} - \frac{\sin re}{0.8333333333333334 \cdot \sin re}\right) \]

    2. metadata-eval 6.1%

       \[\leadsto im \cdot \left(\color{blue}{-0.2} \cdot \frac{\sin re}{\sin re} - \frac{\sin re}{0.8333333333333334 \cdot \sin re}\right) \]

    3. *-inverses 6.1%

       \[\leadsto im \cdot \left(-0.2 \cdot \color{blue}{1} - \frac{\sin re}{0.8333333333333334 \cdot \sin re}\right) \]

    4. metadata-eval 6.1%

       \[\leadsto im \cdot \left(\color{blue}{-0.2} - \frac{\sin re}{0.8333333333333334 \cdot \sin re}\right) \]

    5. *-commutative 6.1%

       \[\leadsto im \cdot \left(-0.2 - \frac{\sin re}{\color{blue}{\sin re \cdot 0.8333333333333334}}\right) \]

    6. associate-/r* 6.1%

       \[\leadsto im \cdot \left(-0.2 - \color{blue}{\frac{\frac{\sin re}{\sin re}}{0.8333333333333334}}\right) \]

    7. *-inverses 6.1%

       \[\leadsto im \cdot \left(-0.2 - \frac{\color{blue}{1}}{0.8333333333333334}\right) \]

    8. metadata-eval 6.1%

       \[\leadsto im \cdot \left(-0.2 - \color{blue}{1.2}\right) \]

    9. metadata-eval 6.1%

       \[\leadsto im \cdot \color{blue}{-1.4} \]

13. Simplified 6.1%

    \[\leadsto im \cdot \color{blue}{-1.4} \]

14. Final simplification 6.1%

    \[\leadsto im \cdot -1.4 \]
15. Add Preprocessing

Alternative 14: 20.3% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 49.3%

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

   1. associate-*r* 49.3%

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

   2. neg-mul-1 49.3%

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

5. Simplified 49.3%

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

   1. pow1 49.3%

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

   2. *-commutative 49.3%

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

   3. add-sqr-sqrt 24.8%

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

   4. sqrt-unprod 38.9%

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

   5. sqr-neg 38.9%

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

   6. sqrt-prod 6.7%

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

   7. add-sqr-sqrt 13.7%

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

7. Applied egg-rr 13.7%

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

   1. unpow1 13.7%

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

   2. *-commutative 13.7%

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

9. Simplified 13.7%

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

10. Taylor expanded in re around 0 18.2%

    \[\leadsto \color{blue}{im \cdot re} \]
11. Step-by-step derivation

    1. *-commutative 18.2%

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

12. Simplified 18.2%

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

13. Final simplification 18.2%

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

Alternative 15: 2.9% accurate, 308.0× speedup

Math

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

FPCore

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

Python

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 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 * 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 * re), $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 re around 0 53.6%

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

   1. associate-*r* 53.6%

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

   2. *-commutative 53.6%

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

5. Simplified 53.6%

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

7. Applied egg-rr 28.9%

   \[\leadsto \color{blue}{\log \left(e^{re}\right)} \]
8. Step-by-step derivation

   1. rem-log-exp 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

    \[\leadsto re \]
11. 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 2024089 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (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))))