math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.5%

Time: 11.6s

Alternatives: 18

Speedup: 2.1×

• 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: 65.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* (sin re) 0.5) t_0)
      (*
       (*
        (fma
         (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
         (* im_m im_m)
         -1.0)
        (sin 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 t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (sin(re) * 0.5) * t_0;
	} else {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * sin(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)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(sin(re) * 0.5) * t_0);
	else
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * sin(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_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   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 95.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot \sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_1, t\_1, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im\_m \cdot im\_m, -1\right) \cdot \sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))))
        (t_1 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_1) t_1 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (if (<= t_0 0.002)
        (* (* (fma -0.16666666666666666 (* im_m im_m) -1.0) (sin re)) im_m)
        (*
         (*
          (fma
           (fma
            (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
            (* re re)
            0.16666666666666666)
           (* re re)
           -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 t_0 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double t_1 = (im_m * im_m) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else if (t_0 <= 0.002) {
		tmp = (fma(-0.16666666666666666, (im_m * im_m), -1.0) * sin(re)) * im_m;
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_1 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(im_m * im_m), -1.0) * sin(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$1), $MachinePrecision] * t$95$1 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.2%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 36.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 99.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 99.7%

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

   if 2e-3 < (*.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

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

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

      6. lower-sin.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 22.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, im \cdot im, -1\right) \cdot \sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_1, t\_1, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot im\_m\right) \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))))
        (t_1 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_1) t_1 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (if (<= t_0 0.002)
        (* (* (sin re) im_m) (fma (* im_m im_m) -0.16666666666666666 -1.0))
        (*
         (*
          (fma
           (fma
            (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
            (* re re)
            0.16666666666666666)
           (* re re)
           -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 t_0 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double t_1 = (im_m * im_m) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else if (t_0 <= 0.002) {
		tmp = (sin(re) * im_m) * fma((im_m * im_m), -0.16666666666666666, -1.0);
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_1 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(sin(re) * im_m) * fma(Float64(im_m * im_m), -0.16666666666666666, -1.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$1), $MachinePrecision] * t$95$1 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[(N[(N[Sin[re], $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.2%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 36.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   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. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

          \[\leadsto \left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{6}, -1\right) \]

      12. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2e-3 < (*.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

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

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

      6. lower-sin.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 22.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_1, t\_1, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))))
        (t_1 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_1) t_1 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (if (<= t_0 0.002)
        (* (- (sin re)) im_m)
        (*
         (*
          (fma
           (fma
            (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
            (* re re)
            0.16666666666666666)
           (* re re)
           -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 t_0 = (sin(re) * 0.5) * (exp(-im_m) - exp(im_m));
	double t_1 = (im_m * im_m) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else if (t_0 <= 0.002) {
		tmp = -sin(re) * im_m;
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_1 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	elseif (t_0 <= 0.002)
		tmp = Float64(Float64(-sin(re)) * im_m);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$1), $MachinePrecision] * t$95$1 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[((-N[Sin[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.2%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2e-3

   1. Initial program 36.8%

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

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

      6. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2e-3 < (*.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

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

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

      6. lower-sin.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 22.1%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0.002:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot t\_0 \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, -0.008333333333333333\right), \left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, -1\right)\right) \cdot \sin re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= (* (* (sin re) 0.5) t_0) (- INFINITY))
      (* (* re 0.5) t_0)
      (*
       (*
        (fma
         (fma (* im_m im_m) -0.0001984126984126984 -0.008333333333333333)
         (* (* (* im_m im_m) im_m) im_m)
         (fma (* im_m im_m) -0.16666666666666666 -1.0))
        (sin 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 t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (((sin(re) * 0.5) * t_0) <= -((double) INFINITY)) {
		tmp = (re * 0.5) * t_0;
	} else {
		tmp = (fma(fma((im_m * im_m), -0.0001984126984126984, -0.008333333333333333), (((im_m * im_m) * im_m) * im_m), fma((im_m * im_m), -0.16666666666666666, -1.0)) * sin(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)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * t_0) <= Float64(-Inf))
		tmp = Float64(Float64(re * 0.5) * t_0);
	else
		tmp = Float64(Float64(fma(fma(Float64(im_m * im_m), -0.0001984126984126984, -0.008333333333333333), Float64(Float64(Float64(im_m * im_m) * im_m) * im_m), fma(Float64(im_m * im_m), -0.16666666666666666, -1.0)) * sin(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_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], (-Infinity)], N[(N[(re * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[re], $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   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 55.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right) \cdot im} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \sin re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right) \cdot im} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), \left(\left(im \cdot im\right) \cdot im\right) \cdot im, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right)\right) \cdot im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), \left(\left(im \cdot im\right) \cdot im\right) \cdot im, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right)\right) \cdot \sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \sin re \cdot 0.5\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_1, t\_1, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (sin re) 0.5)) (t_1 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= (* t_0 (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_1) t_1 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (*
       (*
        (fma
         (fma -0.016666666666666666 (* im_m im_m) -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = sin(re) * 0.5;
	double t_1 = (im_m * im_m) * im_m;
	double tmp;
	if ((t_0 * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else {
		tmp = (fma(fma(-0.016666666666666666, (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * t_0;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(sin(re) * 0.5)
	t_1 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_1), t_1, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	else
		tmp = Float64(Float64(fma(fma(-0.016666666666666666, Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * t_0);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $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[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$1), $MachinePrecision] * t$95$1 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.016666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.2%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   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 55.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_0, t\_0, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot \sin re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))) (- INFINITY))
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_0) t_0 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (*
       (*
        (fma
         (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
         (* im_m im_m)
         -1.0)
        (sin 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 t_0 = (im_m * im_m) * im_m;
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im_m) - exp(im_m))) <= -((double) INFINITY)) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_0), t_0, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * sin(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)
	t_0 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_0), t_0, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	else
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * sin(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_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$0), $MachinePrecision] * t$95$0 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sin[re], $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 re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.2%

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

   10. Applied rewrites 15.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   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 55.4%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 95.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot \sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot t\_0, t\_0, -4.6296296296296296 \cdot 10^{-6}\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* im_m im_m) im_m)))
   (*
    im_s
    (if (<= (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))) 0.0)
      (*
       (*
        (fma
         (fma
          (*
           (/ 1.0 0.0002777777777777778)
           (fma (* -6.248825220858479e-11 t_0) t_0 -4.6296296296296296e-6))
          (* im_m im_m)
          -0.3333333333333333)
         (* im_m im_m)
         -2.0)
        im_m)
       (* re 0.5))
      (*
       (*
        (fma
         (fma
          (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
          (* re re)
          0.16666666666666666)
         (* re re)
         -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 t_0 = (im_m * im_m) * im_m;
	double tmp;
	if (((sin(re) * 0.5) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (fma(fma(((1.0 / 0.0002777777777777778) * fma((-6.248825220858479e-11 * t_0), t_0, -4.6296296296296296e-6)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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)
	t_0 = Float64(Float64(im_m * im_m) * im_m)
	tmp = 0.0
	if (Float64(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(fma(fma(Float64(Float64(1.0 / 0.0002777777777777778) * fma(Float64(-6.248825220858479e-11 * t_0), t_0, -4.6296296296296296e-6)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(1.0 / 0.0002777777777777778), $MachinePrecision] * N[(N[(-6.248825220858479e-11 * t$95$0), $MachinePrecision] * t$95$0 + -4.6296296296296296e-6), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $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))) < 0.0

   1. Initial program 53.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 66.6%

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

   10. Applied rewrites 50.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{\mathsf{fma}\left(1.5747039556563367 \cdot 10^{-7}, \left(im \cdot im\right) \cdot \left(im \cdot im\right), 0.0002777777777777778 - \left(im \cdot im\right) \cdot 6.613756613756614 \cdot 10^{-6}\right)}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(re \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16003008000} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, \frac{-1}{216000}\right) \cdot \frac{1}{\frac{1}{3600}}, im \cdot im, \frac{-1}{3}\right), im \cdot im, -2\right) \cdot im\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 66.6%

       \[\leadsto \left(re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right) \cdot \frac{1}{0.0002777777777777778}, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \]

   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 98.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 21.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{0.0002777777777777778} \cdot \mathsf{fma}\left(-6.248825220858479 \cdot 10^{-11} \cdot \left(\left(im \cdot im\right) \cdot im\right), \left(im \cdot im\right) \cdot im, -4.6296296296296296 \cdot 10^{-6}\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (*
      (fma
       (fma
        (* -0.0003968253968253968 (* im_m im_m))
        (* im_m im_m)
        -0.3333333333333333)
       (* im_m im_m)
       -2.0)
      im_m)
     (* re 0.5))
    (*
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
        (* re re)
        0.16666666666666666)
       (* re re)
       -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 (((sin(re) * 0.5) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (fma(fma((-0.0003968253968253968 * (im_m * im_m)), (im_m * im_m), -0.3333333333333333), (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(fma(fma(Float64(-0.0003968253968253968 * Float64(im_m * im_m)), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $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))) < 0.0

   1. Initial program 53.5%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 66.6%

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

   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 98.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 21.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* (* (sin re) 0.5) (- (exp (- im_m)) (exp im_m))) 0.0)
    (*
     (*
      (fma
       (fma -0.008333333333333333 (* im_m im_m) -0.16666666666666666)
       (* im_m im_m)
       -1.0)
      im_m)
     re)
    (*
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* re re) -0.008333333333333333)
        (* re re)
        0.16666666666666666)
       (* re re)
       -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 (((sin(re) * 0.5) * (exp(-im_m) - exp(im_m))) <= 0.0) {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -0.16666666666666666), (im_m * im_m), -1.0) * im_m) * re;
	} else {
		tmp = (fma(fma(fma(0.0001984126984126984, (re * re), -0.008333333333333333), (re * re), 0.16666666666666666), (re * re), -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(Float64(sin(re) * 0.5) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= 0.0)
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -1.0) * im_m) * re);
	else
		tmp = Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(re * re), -0.008333333333333333), Float64(re * re), 0.16666666666666666), Float64(re * re), -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[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(0.0001984126984126984 * N[(re * re), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $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))) < 0.0

   1. Initial program 53.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 94.5%

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

   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 65.6%

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

   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 98.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{6} + {re}^{2} \cdot \left(\frac{1}{5040} \cdot {re}^{2} - \frac{1}{120}\right)\right) - 1\right)\right) \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 21.4%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sin re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, re \cdot re, -0.008333333333333333\right), re \cdot re, 0.16666666666666666\right), re \cdot re, -1\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im\_m \cdot im\_m, -0.016666666666666666\right), im\_m \cdot im\_m, -0.3333333333333333\right), im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(\sin re \cdot 0.5\right)\right) \end{array} \]

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
     (fma
      (fma -0.0003968253968253968 (* im_m im_m) -0.016666666666666666)
      (* im_m im_m)
      -0.3333333333333333)
     (* im_m im_m)
     -2.0)
    im_m)
   (* (sin re) 0.5))))

C

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

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(fma(fma(-0.0003968253968253968, Float64(im_m * im_m), -0.016666666666666666), Float64(im_m * im_m), -0.3333333333333333), Float64(im_m * im_m), -2.0) * im_m) * Float64(sin(re) * 0.5)))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in im around 0

   \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
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 95.9%

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

6. Final simplification 95.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right) \cdot \left(\sin re \cdot 0.5\right) \]
7. Add Preprocessing

Alternative 12: 56.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (sin re) -0.07)
    (*
     (* (fma -0.3333333333333333 (* im_m im_m) -2.0) im_m)
     (* (fma (* re re) -0.08333333333333333 0.5) re))
    (*
     (*
      (fma
       (fma -0.008333333333333333 (* im_m im_m) -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 (sin(re) <= -0.07) {
		tmp = (fma(-0.3333333333333333, (im_m * im_m), -2.0) * im_m) * (fma((re * re), -0.08333333333333333, 0.5) * re);
	} else {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -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 (sin(re) <= -0.07)
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * im_m) * Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re));
	else
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -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[Sin[re], $MachinePrecision], -0.07], N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.070000000000000007

   1. Initial program 52.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   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. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.1

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

   8. Applied rewrites 21.1%

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

   if -0.070000000000000007 < (sin.f64 re)

   1. Initial program 67.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.5%

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

   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 76.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im\_m \cdot im\_m, -0.16666666666666666\right), im\_m \cdot im\_m, -1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (sin re) -0.07)
    (* (* (fma 0.16666666666666666 (* re re) -1.0) im_m) re)
    (*
     (*
      (fma
       (fma -0.008333333333333333 (* im_m im_m) -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 (sin(re) <= -0.07) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * im_m) * re;
	} else {
		tmp = (fma(fma(-0.008333333333333333, (im_m * im_m), -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 (sin(re) <= -0.07)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * im_m) * re);
	else
		tmp = Float64(Float64(fma(fma(-0.008333333333333333, Float64(im_m * im_m), -0.16666666666666666), Float64(im_m * im_m), -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[Sin[re], $MachinePrecision], -0.07], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.070000000000000007

   1. Initial program 52.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   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 21.1%

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

   if -0.070000000000000007 < (sin.f64 re)

   1. Initial program 67.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   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 91.5%

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

   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 76.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im\_m \cdot im\_m, -2\right) \cdot im\_m\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (sin re) -0.07)
    (* (* (fma 0.16666666666666666 (* re re) -1.0) im_m) re)
    (* (* (fma -0.3333333333333333 (* im_m im_m) -2.0) im_m) (* re 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (sin(re) <= -0.07) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * im_m) * re;
	} else {
		tmp = (fma(-0.3333333333333333, (im_m * im_m), -2.0) * im_m) * (re * 0.5);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (sin(re) <= -0.07)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * im_m) * re);
	else
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(im_m * im_m), -2.0) * im_m) * Float64(re * 0.5));
	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[Sin[re], $MachinePrecision], -0.07], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision] + -2.0), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.070000000000000007

   1. Initial program 52.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   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 21.1%

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

   if -0.070000000000000007 < (sin.f64 re)

   1. Initial program 67.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   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. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-re\right) \cdot re}{re} \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (sin re) 1e-5)
    (* (* (fma 0.16666666666666666 (* re re) -1.0) im_m) re)
    (* (/ (* (- re) 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 (sin(re) <= 1e-5) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * im_m) * re;
	} else {
		tmp = ((-re * re) / 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 (sin(re) <= 1e-5)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * im_m) * re);
	else
		tmp = Float64(Float64(Float64(Float64(-re) * re) / 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[Sin[re], $MachinePrecision], 1e-5], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[((-re) * re), $MachinePrecision] / re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000008e-5

   1. Initial program 68.3%

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

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

      6. lower-sin.f64 58.1

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

   5. Applied rewrites 58.1%

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

   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 49.5%

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

   if 1.00000000000000008e-5 < (sin.f64 re)

   1. Initial program 49.4%

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

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

      6. lower-sin.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.0%

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

   10. Applied rewrites 6.9%

       \[\leadsto \frac{0 - \left(re \cdot re\right) \cdot re}{0 + \mathsf{fma}\left(re, re, 0 \cdot re\right)} \cdot im \]
   11. Step-by-step derivation

   12. Applied rewrites 17.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-re\right) \cdot re}{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, re \cdot re, -1\right) \cdot im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (sin re) 1e-5)
    (* (* (fma 0.16666666666666666 (* re re) -1.0) 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 (sin(re) <= 1e-5) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * 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 (sin(re) <= 1e-5)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * im_m) * re);
	else
		tmp = Float64(Float64(-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[Sin[re], $MachinePrecision], 1e-5], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < 1.00000000000000008e-5

   1. Initial program 68.3%

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

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

      6. lower-sin.f64 58.1

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

   5. Applied rewrites 58.1%

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

   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 49.5%

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

   if 1.00000000000000008e-5 < (sin.f64 re)

   1. Initial program 49.4%

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

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

      6. lower-sin.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.0%

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

4. Final simplification 42.7%

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

Alternative 17: 34.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\left(\left(re \cdot im\_m\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (sin re) -0.07)
    (* (* (* (* re im_m) re) 0.16666666666666666) 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 (sin(re) <= -0.07) {
		tmp = (((re * im_m) * re) * 0.16666666666666666) * 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 (sin(re) <= (-0.07d0)) then
        tmp = (((re * im_m) * re) * 0.16666666666666666d0) * 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 (Math.sin(re) <= -0.07) {
		tmp = (((re * im_m) * re) * 0.16666666666666666) * 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 math.sin(re) <= -0.07:
		tmp = (((re * im_m) * re) * 0.16666666666666666) * 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 (sin(re) <= -0.07)
		tmp = Float64(Float64(Float64(Float64(re * im_m) * re) * 0.16666666666666666) * 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 (sin(re) <= -0.07)
		tmp = (((re * im_m) * re) * 0.16666666666666666) * 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[Sin[re], $MachinePrecision], -0.07], N[(N[(N[(N[(re * im$95$m), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 re) < -0.070000000000000007

   1. Initial program 52.7%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   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 21.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 20.4%

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

   if -0.070000000000000007 < (sin.f64 re)

   1. Initial program 67.5%

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

   3. Taylor expanded in im around 0

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

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

      6. lower-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 48.2%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin re \leq -0.07:\\ \;\;\;\;\left(\left(\left(re \cdot im\right) \cdot re\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.7% 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 64.4%

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

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

   6. lower-sin.f64 57.6

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 40.9%

   \[\leadsto \left(-re\right) \cdot im \]
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 2024240 
(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))))