math.cos on complex, imaginary part

Percentage Accurate: 66.0% → 99.9%

Time: 10.3s

Alternatives: 22

Speedup: 1.4×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0013:\\ \;\;\;\;t\_0 \cdot \left(\left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= im_m 0.0013)
      (* t_0 (* (- (* -0.3333333333333333 (* im_m im_m)) 2.0) im_m))
      (* t_0 (- (exp (- im_m)) (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im_m <= 0.0013) {
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * (exp(-im_m) - exp(im_m));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im_m <= 0.0013d0) then
        tmp = t_0 * ((((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0) * im_m)
    else
        tmp = t_0 * (exp(-im_m) - exp(im_m))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im_m <= 0.0013) {
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * (Math.exp(-im_m) - Math.exp(im_m));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im_m <= 0.0013:
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m)
	else:
		tmp = t_0 * (math.exp(-im_m) - math.exp(im_m))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im_m <= 0.0013)
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im_m <= 0.0013)
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	else
		tmp = t_0 * (exp(-im_m) - exp(im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 0.0013], N[(t$95$0 * N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0012999999999999999

   1. Initial program 55.3%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)} \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\color{blue}{\frac{-1}{3} \cdot {im}^{2}} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

      6. lower-*.f64 89.5

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

   5. Applied rewrites 89.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 0.0012999999999999999 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (if (<= t_0 2e-10)
        (* (- (sin re)) im_m)
        (*
         (* (fma (* re re) -0.08333333333333333 0.5) re)
         (*
          (-
           (*
            (-
             (*
              (*
               (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else if (t_0 <= 2e-10) {
		tmp = -sin(re) * im_m;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	elseif (t_0 <= 2e-10)
		tmp = Float64(Float64(-sin(re)) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * 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[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], N[((-N[Sin[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2.00000000000000007e-10

   1. Initial program 34.8%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 2.00000000000000007e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lower-*.f64 76.0

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_1 := \left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot t\_1\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))))
        (t_1
         (*
          (-
           (*
            (-
             (*
              (*
               (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
               im_m)
              im_m)
             0.3333333333333333)
            (* im_m im_m))
           2.0)
          im_m)))
   (*
    im_s
    (if (<= t_0 -2e-186)
      (*
       (*
        (fma
         (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
         (* re re)
         0.5)
        re)
       t_1)
      (if (<= t_0 2e-10)
        (* (- (sin re)) im_m)
        (* (* (fma (* re re) -0.08333333333333333 0.5) re) t_1))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double t_1 = (((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m;
	double tmp;
	if (t_0 <= -2e-186) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * t_1;
	} else if (t_0 <= 2e-10) {
		tmp = -sin(re) * im_m;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * t_1;
	}
	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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m)
	tmp = 0.0
	if (t_0 <= -2e-186)
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * t_1);
	elseif (t_0 <= 2e-10)
		tmp = Float64(Float64(-sin(re)) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * t_1);
	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[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -2e-186], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e-10], N[((-N[Sin[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * t$95$1), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1.9999999999999998e-186

   1. Initial program 99.1%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 86.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 86.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) + \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. lower--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{1}{240} \cdot {re}^{2}} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       8. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{12}, {re}^{2}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       11. lower-*.f64 73.3

           \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 73.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if -1.9999999999999998e-186 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2.00000000000000007e-10

   1. Initial program 34.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 2.00000000000000007e-10 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lower-*.f64 76.0

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = t_0 * ((((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 97.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 97.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (* (* 0.5 re) (- 1.0 (exp im_m)))
    (*
     (*
      (sin re)
      (fma
       (* im_m im_m)
       (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
       -1.0))
     im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = (sin(re) * fma((im_m * im_m), fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666), -1.0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(sin(re) * fma(Float64(im_m * im_m), fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), -1.0)) * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (* (* 0.5 re) (- 1.0 (exp im_m)))
    (*
     (*
      (sin re)
      (fma (* im_m im_m) (* (* im_m im_m) -0.008333333333333333) -1.0))
     im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = (sin(re) * fma((im_m * im_m), ((im_m * im_m) * -0.008333333333333333), -1.0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(sin(re) * fma(Float64(im_m * im_m), Float64(Float64(im_m * im_m) * -0.008333333333333333), -1.0)) * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in im around inf

      \[\leadsto \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{120} \cdot {im}^{2}, -1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 94.3%

      \[\leadsto \left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \left(im \cdot im\right) \cdot -0.008333333333333333, -1\right)\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (* (* 0.5 re) (- 1.0 (exp im_m)))
      (* t_0 (* (- (* -0.3333333333333333 (* im_m im_m)) 2.0) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	} else {
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * 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 = 0.5 * Math.sin(re);
	double tmp;
	if ((t_0 * (Math.exp(-im_m) - Math.exp(im_m))) <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * re) * (1.0 - Math.exp(im_m));
	} else {
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if (t_0 * (math.exp(-im_m) - math.exp(im_m))) <= -math.inf:
		tmp = (0.5 * re) * (1.0 - math.exp(im_m))
	else:
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0) * im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -Inf)
		tmp = (0.5 * re) * (1.0 - exp(im_m));
	else
		tmp = t_0 * (((-0.3333333333333333 * (im_m * im_m)) - 2.0) * im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)} \cdot im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\color{blue}{\frac{-1}{3} \cdot {im}^{2}} - 2\right) \cdot im\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(\frac{-1}{3} \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

      6. lower-*.f64 90.4

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(-0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

   5. Applied rewrites 90.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.1% accurate, 0.7× speedup

Math

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

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 (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
    (* (* 0.5 re) (- 1.0 (exp im_m)))
    (* (* (sin re) im_m) (fma (* -0.16666666666666666 im_m) im_m -1.0)))))

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(sin(re) * im_m) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0));
	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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 39.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

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

      1. lower-*.f64 31.3

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right) \cdot re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, im\_m, 1\right), im\_m, 1\right)\right)\\ \end{array} \end{array} \]

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 (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (*
      (-
       (*
        (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
        im_m)
       1.0)
      re)
     im_m)
    (*
     (* (fma (* re re) -0.08333333333333333 0.5) re)
     (- 1.0 (fma (fma 0.5 im_m 1.0) im_m 1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = ((((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * im_m;
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (1.0 - fma(fma(0.5, im_m, 1.0), im_m, 1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(1.0 - fma(fma(0.5, im_m, 1.0), im_m, 1.0)));
	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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * im$95$m + 1.0), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 98.8%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 60.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(1 - e^{im}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(1 - e^{im}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - e^{im}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(1 - e^{im}\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - e^{im}\right) \]

      7. lower-*.f64 55.3

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - e^{im}\right) \]

   8. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(1 - e^{im}\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right) + 1\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - \left(\color{blue}{\left(1 + \frac{1}{2} \cdot im\right) \cdot im} + 1\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - \color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot im, im, 1\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(1 - \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot im + 1}, im, 1\right)\right) \]

       5. lower-fma.f64 41.7

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, im, 1\right)}, im, 1\right)\right) \]

   11. Applied rewrites 41.7%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -0.0002:\\ \;\;\;\;\left(\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) -0.0002)
    (* (* (* (* im_m im_m) -0.16666666666666666) im_m) re)
    (* (* re (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -0.0002) {
		tmp = (((im_m * im_m) * -0.16666666666666666) * im_m) * re;
	} else {
		tmp = (re * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -0.0002)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) * im_m) * re);
	else
		tmp = Float64(Float64(re * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * 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_] := N[(im$95$s * If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(re * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in im around inf

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 59.1%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot re \]

   if -2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 57.5%

      \[\leadsto \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.6%

       \[\leadsto \color{blue}{\left(re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 43.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -0.0002:\\ \;\;\;\;\left(\left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) -0.0002)
    (* (* (* (* im_m im_m) -0.16666666666666666) im_m) re)
    (* (- re) 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 (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -0.0002) {
		tmp = (((im_m * im_m) * -0.16666666666666666) * im_m) * re;
	} else {
		tmp = -re * 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 (((0.5d0 * sin(re)) * (exp(-im_m) - exp(im_m))) <= (-0.0002d0)) then
        tmp = (((im_m * im_m) * (-0.16666666666666666d0)) * im_m) * re
    else
        tmp = -re * 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 (((0.5 * Math.sin(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= -0.0002) {
		tmp = (((im_m * im_m) * -0.16666666666666666) * im_m) * re;
	} else {
		tmp = -re * 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 ((0.5 * math.sin(re)) * (math.exp(-im_m) - math.exp(im_m))) <= -0.0002:
		tmp = (((im_m * im_m) * -0.16666666666666666) * im_m) * re
	else:
		tmp = -re * 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 (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -0.0002)
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * -0.16666666666666666) * im_m) * re);
	else
		tmp = Float64(Float64(-re) * 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 (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -0.0002)
		tmp = (((im_m * im_m) * -0.16666666666666666) * im_m) * re;
	else
		tmp = -re * 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[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2.0000000000000001e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in im around inf

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot im\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 59.1%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right) \cdot re \]

   if -2.0000000000000001e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 55.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.4%

      \[\leadsto \left(-re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7:\\ \;\;\;\;t\_0 \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - e^{im\_m}\right)\\ \end{array} \end{array} \end{array} \]

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 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= im_m 3.7)
      (*
       t_0
       (*
        (-
         (*
          (-
           (*
            (*
             (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
             im_m)
            im_m)
           0.3333333333333333)
          (* im_m im_m))
         2.0)
        im_m))
      (* t_0 (- 1.0 (exp im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im_m <= 3.7) {
		tmp = t_0 * ((((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = t_0 * (1.0 - exp(im_m));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im_m <= 3.7)
		tmp = Float64(t_0 * Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(t_0 * Float64(1.0 - exp(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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.7], N[(t$95$0 * N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7000000000000002

   1. Initial program 55.6%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 96.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if 3.7000000000000002 < 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

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

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 59.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot im\_m, re \cdot re, im\_m\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) 2e-13)
    (*
     (* (fma (* re re) -0.08333333333333333 0.5) re)
     (*
      (-
       (*
        (-
         (*
          (*
           (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
           im_m)
          im_m)
         0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m))
    (*
     (*
      (fma
       (* (fma 0.008333333333333333 (* re re) -0.16666666666666666) im_m)
       (* re re)
       im_m)
      re)
     (fma (* -0.16666666666666666 im_m) im_m -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-13) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * ((((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	} else {
		tmp = (fma((fma(0.008333333333333333, (re * re), -0.16666666666666666) * im_m), (re * re), im_m) * re) * fma((-0.16666666666666666 * im_m), im_m, -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-13)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * im_m));
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(re * re), -0.16666666666666666) * im_m), Float64(re * re), im_m) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0));
	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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + im$95$m), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2.0000000000000001e-13

   1. Initial program 71.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 95.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 95.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

       7. lower-*.f64 74.3

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 74.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   if 2.0000000000000001e-13 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 45.3%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + {re}^{2} \cdot \left(\frac{-1}{6} \cdot im + \frac{1}{120} \cdot \left(im \cdot {re}^{2}\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot im}, im, -1\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + {re}^{2} \cdot \left(\frac{-1}{6} \cdot im + \frac{1}{120} \cdot \left(im \cdot {re}^{2}\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot im, re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot im}, im, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right)\right) \cdot re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right) \cdot im\_m\right) \cdot im\_m - 0.3333333333333333\right) \cdot \left(im\_m \cdot im\_m\right) - 2\right) \cdot im\_m\right)\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (*
     (*
      (*
       (fma -0.16666666666666666 (* re re) 1.0)
       (-
        (*
         (*
          (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
          im_m)
         im_m)
        1.0))
      re)
     im_m)
    (*
     (* 0.5 re)
     (*
      (-
       (*
        (-
         (*
          (*
           (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
           im_m)
          im_m)
         0.3333333333333333)
        (* im_m im_m))
       2.0)
      im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((fma(-0.16666666666666666, (re * re), 1.0) * (((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0)) * re) * im_m;
	} else {
		tmp = (0.5 * re) * ((((((fma(-0.0003968253968253968, (im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * (im_m * im_m)) - 2.0) * im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0)) * re) * im_m);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(Float64(Float64(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666) * im_m) * im_m) - 0.3333333333333333) * Float64(im_m * im_m)) - 2.0) * 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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + {im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) - 1\right)\right) \cdot im \]

   8. Applied rewrites 22.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   6. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   8. Applied rewrites 94.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(\frac{-1}{2520}, im \cdot im, \frac{-1}{60}\right) \cdot im\right) \cdot im - \frac{1}{3}\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 78.2

          \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]

   11. Applied rewrites 78.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(\left(\left(\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right)\right) \cdot re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot im\_m, re \cdot re, im\_m\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) 2e-13)
    (*
     (*
      (*
       (fma -0.16666666666666666 (* re re) 1.0)
       (-
        (*
         (*
          (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666)
          im_m)
         im_m)
        1.0))
      re)
     im_m)
    (*
     (*
      (fma
       (* (fma 0.008333333333333333 (* re re) -0.16666666666666666) im_m)
       (* re re)
       im_m)
      re)
     (fma (* -0.16666666666666666 im_m) im_m -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-13) {
		tmp = ((fma(-0.16666666666666666, (re * re), 1.0) * (((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0)) * re) * im_m;
	} else {
		tmp = (fma((fma(0.008333333333333333, (re * re), -0.16666666666666666) * im_m), (re * re), im_m) * re) * fma((-0.16666666666666666 * im_m), im_m, -1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-13)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0)) * re) * im_m);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.008333333333333333, Float64(re * re), -0.16666666666666666) * im_m), Float64(re * re), im_m) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0));
	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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + im$95$m), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 2.0000000000000001e-13

   1. Initial program 71.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(\left(\frac{-1}{6} \cdot \left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) + {im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) - 1\right)\right) \cdot im \]

   8. Applied rewrites 69.6%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right)\right) \cdot re\right) \cdot im \]

   if 2.0000000000000001e-13 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 45.3%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + {re}^{2} \cdot \left(\frac{-1}{6} \cdot im + \frac{1}{120} \cdot \left(im \cdot {re}^{2}\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot im}, im, -1\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + {re}^{2} \cdot \left(\frac{-1}{6} \cdot im + \frac{1}{120} \cdot \left(im \cdot {re}^{2}\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, re \cdot re, -0.16666666666666666\right) \cdot im, re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot im}, im, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(im\_m \cdot re\right) \cdot re, -0.16666666666666666, im\_m\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* 0.5 (sin re)) -0.01)
    (*
     (* (fma (* (* im_m re) re) -0.16666666666666666 im_m) re)
     (fma (* -0.16666666666666666 im_m) im_m -1.0))
    (*
     (*
      (-
       (*
        (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
        im_m)
       1.0)
      re)
     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 ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(((im_m * re) * re), -0.16666666666666666, im_m) * re) * fma((-0.16666666666666666 * im_m), im_m, -1.0);
	} else {
		tmp = ((((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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 (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(Float64(im_m * re) * re), -0.16666666666666666, im_m) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(N[(im$95$m * re), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + im$95$m), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + \frac{-1}{6} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

      \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 57.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, re \cdot re, im\_m\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* 0.5 (sin re)) -0.01)
    (*
     (* (fma (* -0.16666666666666666 im_m) (* re re) im_m) re)
     (fma (* -0.16666666666666666 im_m) im_m -1.0))
    (*
     (*
      (-
       (*
        (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
        im_m)
       1.0)
      re)
     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 ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((-0.16666666666666666 * im_m), (re * re), im_m) * re) * fma((-0.16666666666666666 * im_m), im_m, -1.0);
	} else {
		tmp = ((((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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 (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * im_m), Float64(re * re), im_m) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * N[(re * re), $MachinePrecision] + im$95$m), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left(im + {re}^{2} \cdot \left(\frac{-1}{6} \cdot im + \frac{1}{120} \cdot \left(im \cdot {re}^{2}\right)\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot im}, im, -1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.008333333333333333, -0.16666666666666666 \cdot im\right), re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot im}, im, -1\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{6} \cdot im, re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot im, im, -1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 21.2%

       \[\leadsto \left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, re \cdot re, im\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right) \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

      \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 56.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im\_m \cdot re\right) \cdot re, 0.16666666666666666, -im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* 0.5 (sin re)) -0.01)
    (* (fma (* (* im_m re) re) 0.16666666666666666 (- im_m)) re)
    (*
     (*
      (-
       (*
        (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
        im_m)
       1.0)
      re)
     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 ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im_m * re) * re), 0.16666666666666666, -im_m) * re;
	} else {
		tmp = ((((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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 (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im_m * re) * re), 0.16666666666666666, Float64(-im_m)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * re) * 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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im$95$m * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

      \[\leadsto \left(\left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 56.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im\_m \cdot re\right) \cdot re, 0.16666666666666666, -im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.008333333333333333, -0.16666666666666666\right) \cdot im\_m\right) \cdot im\_m - 1\right) \cdot \left(im\_m \cdot re\right)\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (fma (* (* im_m re) re) 0.16666666666666666 (- im_m)) re)
    (*
     (-
      (*
       (* (fma (* im_m im_m) -0.008333333333333333 -0.16666666666666666) im_m)
       im_m)
      1.0)
     (* im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im_m * re) * re), 0.16666666666666666, -im_m) * re;
	} else {
		tmp = (((fma((im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * (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 (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im_m * re) * re), 0.16666666666666666, Float64(-im_m)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(im_m * im_m), -0.008333333333333333, -0.16666666666666666) * im_m) * im_m) - 1.0) * Float64(im_m * re));
	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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im$95$m * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) \cdot im} \]

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]

   6. Taylor expanded in re around 0

      \[\leadsto im \cdot \color{blue}{\left(re \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{\left(im \cdot re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 53.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im\_m \cdot re\right) \cdot re, 0.16666666666666666, -im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, -1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (fma (* (* im_m re) re) 0.16666666666666666 (- im_m)) re)
    (* (* (fma (* im_m im_m) -0.16666666666666666 -1.0) im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im_m * re) * re), 0.16666666666666666, -im_m) * re;
	} else {
		tmp = (fma((im_m * im_m), -0.16666666666666666, -1.0) * 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 (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im_m * re) * re), 0.16666666666666666, Float64(-im_m)) * re);
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), -0.16666666666666666, -1.0) * im_m) * re);
	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_] := N[(im$95$s * If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im$95$m * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sin.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 69.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

      16. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.1%

      \[\leadsto \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 54.1% accurate, 14.4× speedup

Math

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

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 (* (* (fma (* im_m im_m) -0.16666666666666666 -1.0) im_m) re)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((fma((im_m * im_m), -0.16666666666666666, -1.0) * 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(Float64(fma(Float64(im_m * im_m), -0.16666666666666666, -1.0) * 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[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in im around 0

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

   1. +-commutative N/A

      \[\leadsto im \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right) + -1 \cdot \sin re\right)} \]

   2. *-commutative N/A

      \[\leadsto im \cdot \left(\color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}} + -1 \cdot \sin re\right) \]

   3. associate-*r* N/A

      \[\leadsto im \cdot \left(\color{blue}{{im}^{2} \cdot \left(\sin re \cdot \frac{-1}{6}\right)} + -1 \cdot \sin re\right) \]

   4. *-commutative N/A

      \[\leadsto im \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \sin re\right)} + -1 \cdot \sin re\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) + im \cdot \left(-1 \cdot \sin re\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re\right)\right) \cdot im} + im \cdot \left(-1 \cdot \sin re\right) \]

   7. *-commutative N/A

      \[\leadsto \left({im}^{2} \cdot \color{blue}{\left(\sin re \cdot \frac{-1}{6}\right)}\right) \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{-1}{6}\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \sin re\right)\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \sin re\right)} \cdot im + im \cdot \left(-1 \cdot \sin re\right) \]

   11. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(\sin re \cdot im\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot \sin re\right)} + im \cdot \left(-1 \cdot \sin re\right) \]

   13. mul-1-neg N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + im \cdot \color{blue}{\left(\mathsf{neg}\left(\sin re\right)\right)} \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{\left(\mathsf{neg}\left(im \cdot \sin re\right)\right)} \]

   15. mul-1-neg N/A

       \[\leadsto \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \left(im \cdot \sin re\right) + \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]

   16. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\left(im \cdot \sin re\right) \cdot \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)} \]

5. Applied rewrites 82.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 57.9%

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

Alternative 22: 33.5% accurate, 39.5× speedup

Math

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

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 (* (- re) im_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 35.9%

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

Developer Target 1: 99.8% accurate, 1.0× 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 2024339 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

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

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