math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 98.1%

Time: 21.4s

Alternatives: 24

Speedup: 1.0×

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot im\_m\\ t_1 := e^{re} \cdot \sin im\_m\\ t_2 := \sin im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot \left(-0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (exp re) im_m))
        (t_1 (* (exp re) (sin im_m)))
        (t_2 (* (sin im_m) (fma re (fma re 0.5 1.0) 1.0))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_1 -0.02)
        t_2
        (if (<= t_1 0.0) t_0 (if (<= t_1 1.0) t_2 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 = exp(re) * im_m;
	double t_1 = exp(re) * sin(im_m);
	double t_2 = sin(im_m) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_1 <= -0.02) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = 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(exp(re) * im_m)
	t_1 = Float64(exp(re) * sin(im_m))
	t_2 = Float64(sin(im_m) * fma(re, fma(re, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_1 <= -0.02)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = 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[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im$95$m], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$2, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 38.5

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 38.5

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 38.5%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 95.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := \sin im\_m \cdot \left(re + 1\right)\\ t_2 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot \left(-0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 re) (sin im_m)))
        (t_1 (* (sin im_m) (+ re 1.0)))
        (t_2 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_0 -0.02)
        t_1
        (if (<= t_0 0.0) t_2 (if (<= t_0 1.0) t_1 t_2)))))))

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(re) * sin(im_m);
	double t_1 = sin(im_m) * (re + 1.0);
	double t_2 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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(re) * sin(im_m))
	t_1 = Float64(sin(im_m) * Float64(re + 1.0))
	t_2 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_0 <= -0.02)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1.0)
		tmp = t_1;
	else
		tmp = t_2;
	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[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im$95$m], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], t$95$1, If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1.0], t$95$1, t$95$2]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 38.5

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 38.5

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 38.5%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.3

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

   5. Simplified 99.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 95.5%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ t_1 := e^{re} \cdot im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot \left(-0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))) (t_1 (* (exp re) im_m)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_0 -0.02)
        (sin im_m)
        (if (<= t_0 2e-13) t_1 (if (<= t_0 1.0) (sin im_m) t_1)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double t_1 = exp(re) * im_m;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = sin(im_m);
	} else if (t_0 <= 2e-13) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = t_1;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	t_1 = Float64(exp(re) * im_m)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_0 <= -0.02)
		tmp = sin(im_m);
	elseif (t_0 <= 2e-13)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = t_1;
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im$95$m], $MachinePrecision], If[LessEqual[t$95$0, 2e-13], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im$95$m], $MachinePrecision], t$95$1]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 38.5

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 38.5

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 38.5%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 2.0000000000000001e-13 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 96.0

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

   5. Simplified 96.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-13 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 96.3%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot \left(-0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\_m\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_0 -0.02)
        (sin im_m)
        (if (<= t_0 0.0)
          (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
          (if (<= t_0 1.0)
            (sin im_m)
            (*
             im_m
             (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = sin(im_m);
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_0 <= -0.02)
		tmp = sin(im_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	elseif (t_0 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im$95$m], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im$95$m], $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 38.5

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 38.5

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 38.5%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 97.2

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

   5. Simplified 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 80.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 62.2

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 62.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.4× speedup

Math

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

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 re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_0 1.0)
        (/
         (sin im_m)
         (+ 1.0 (* (* re (fma re 0.5 1.0)) (fma re (fma re 0.5 1.0) -1.0))))
        (* (exp 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(re) * sin(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(im_m) / (1.0 + ((re * fma(re, 0.5, 1.0)) * fma(re, fma(re, 0.5, 1.0), -1.0)));
	} else {
		tmp = exp(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(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im_m) / Float64(1.0 + Float64(Float64(re * fma(re, 0.5, 1.0)) * fma(re, fma(re, 0.5, 1.0), -1.0))));
	else
		tmp = Float64(exp(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[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[im$95$m], $MachinePrecision] / N[(1.0 + N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 2.7

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

   5. Simplified 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 38.5

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 38.5%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 38.5

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 38.5%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 73.2

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

   5. Simplified 73.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 72.8%

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

   8. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 97.2

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

   10. Simplified 97.2%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 80.6%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \mathsf{fma}\left(re \cdot re, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (fma
       im_m
       (fma
        (* im_m im_m)
        (*
         (+ re 1.0)
         (fma
          im_m
          (*
           im_m
           (fma (* im_m im_m) -0.0001984126984126984 0.008333333333333333))
          -0.16666666666666666))
        re)
       im_m)
      (if (<= t_0 0.0)
        (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
        (*
         (fma (* re (fma (* re re) 0.25 -1.0)) (/ 1.0 (fma re 0.5 -1.0)) 1.0)
         (*
          im_m
          (fma
           (* im_m im_m)
           (fma (* im_m im_m) 0.008333333333333333 -0.16666666666666666)
           1.0))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(im_m, fma((im_m * im_m), ((re + 1.0) * fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re), im_m);
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = fma((re * fma((re * re), 0.25, -1.0)), (1.0 / fma(re, 0.5, -1.0)), 1.0) * (im_m * fma((im_m * im_m), fma((im_m * im_m), 0.008333333333333333, -0.16666666666666666), 1.0));
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = fma(im_m, fma(Float64(im_m * im_m), Float64(Float64(re + 1.0) * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re), im_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(fma(Float64(re * fma(Float64(re * re), 0.25, -1.0)), Float64(1.0 / fma(re, 0.5, -1.0)), 1.0) * Float64(im_m * fma(Float64(im_m * im_m), fma(Float64(im_m * im_m), 0.008333333333333333, -0.16666666666666666), 1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] + im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * N[(N[(re * re), $MachinePrecision] * 0.25 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(re * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 19.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(1 + re\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\right)} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{2}, 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      11. *-lowering-*.f64 50.6

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]

   8. Simplified 50.6%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(re \cdot \frac{1}{2}\right) \cdot \left(re \cdot \frac{1}{2}\right) - 1 \cdot 1\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(re \cdot \frac{1}{2}\right) \cdot \left(re \cdot \frac{1}{2}\right) - 1 \cdot 1\right) \cdot re}, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(re \cdot \frac{1}{2}\right) \cdot \left(re \cdot \frac{1}{2}\right) - \color{blue}{1}\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(re \cdot \frac{1}{2}\right) \cdot \left(re \cdot \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(re \cdot re\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(re \cdot re, \frac{1}{2} \cdot \frac{1}{2}, -1\right)} \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{1}{2} \cdot \frac{1}{2}, -1\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \color{blue}{\frac{1}{4}}, -1\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} - 1}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{4}, -1\right) \cdot re, \color{blue}{\frac{1}{re \cdot \frac{1}{2} - 1}}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{4}, -1\right) \cdot re, \frac{1}{\color{blue}{re \cdot \frac{1}{2} + \left(\mathsf{neg}\left(1\right)\right)}}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{4}, -1\right) \cdot re, \frac{1}{re \cdot \frac{1}{2} + \color{blue}{-1}}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      17. accelerator-lowering-fma.f64 54.5

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.25, -1\right) \cdot re, \frac{1}{\color{blue}{\mathsf{fma}\left(re, 0.5, -1\right)}}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]

   10. Applied egg-rr 54.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.25, -1\right) \cdot re, \frac{1}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right)} \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(re + 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot \mathsf{fma}\left(re \cdot re, 0.25, -1\right), \frac{1}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im\_m, \mathsf{fma}\left(im\_m \cdot im\_m, \left(re + 1\right) \cdot \mathsf{fma}\left(im\_m, im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (fma
       im_m
       (fma
        (* im_m im_m)
        (*
         (+ re 1.0)
         (fma
          im_m
          (*
           im_m
           (fma (* im_m im_m) -0.0001984126984126984 0.008333333333333333))
          -0.16666666666666666))
        re)
       im_m)
      (if (<= t_0 0.0)
        (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
        (*
         im_m
         (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(im_m, fma((im_m * im_m), ((re + 1.0) * fma(im_m, (im_m * fma((im_m * im_m), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re), im_m);
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = fma(im_m, fma(Float64(im_m * im_m), Float64(Float64(re + 1.0) * fma(im_m, Float64(im_m * fma(Float64(im_m * im_m), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666)), re), im_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(re + 1.0), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + re), $MachinePrecision] + im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 19.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(1 + re\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\right)} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 54.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, \mathsf{fma}\left(im \cdot im, \left(re + 1\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), re\right), im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot \left(-0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (*
       im_m
       (fma
        (* im_m im_m)
        (* im_m (* -0.0001984126984126984 (* im_m (* im_m im_m))))
        1.0))
      (if (<= t_0 0.0)
        (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
        (*
         im_m
         (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = im_m * fma((im_m * im_m), (im_m * (-0.0001984126984126984 * (im_m * (im_m * im_m)))), 1.0);
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(im_m * fma(Float64(im_m * im_m), Float64(im_m * Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * im_m)))), 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 55.9

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

   5. Simplified 55.9%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. associate-*l* N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      9. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right) \]

      18. *-lowering-*.f64 19.3

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 19.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{4}}, 1\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right) \]

       2. pow-sqr N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{5040} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}, 1\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2}}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)}, 1\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right), 1\right) \]

       6. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2}\right)\right)}, 1\right) \]

       7. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\left(\frac{-1}{5040} \cdot {im}^{2}\right) \cdot im\right)}, 1\right) \]

       8. associate-*r* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot \left({im}^{2} \cdot im\right)\right)}, 1\right) \]

       9. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot im\right)\right), 1\right) \]

       10. unpow3 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{{im}^{3}}\right), 1\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{3}\right)}, 1\right) \]

       13. cube-mult N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot \left(im \cdot im\right)\right)}\right), 1\right) \]

       14. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{{im}^{2}}\right)\right), 1\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \color{blue}{\left(im \cdot {im}^{2}\right)}\right), 1\right) \]

       16. unpow2 N/A

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(\frac{-1}{5040} \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

       17. *-lowering-*.f64 18.2

           \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \color{blue}{\left(im \cdot im\right)}\right)\right), 1\right) \]

   11. Simplified 18.2%

       \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)}, 1\right) \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 54.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, im \cdot \left(-0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;im\_m \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (*
       im_m
       (*
        (fma re (fma re 0.5 1.0) 1.0)
        (fma im_m (* im_m -0.16666666666666666) 1.0)))
      (if (<= t_0 0.0)
        (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
        (*
         im_m
         (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = im_m * (fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im_m, (im_m * -0.16666666666666666), 1.0));
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(im_m * Float64(fma(re, fma(re, 0.5, 1.0), 1.0) * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(im$95$m * N[(N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 83.0

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

   5. Simplified 83.0%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 27.7%

      \[\leadsto \color{blue}{im \cdot \left(\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\right)} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 54.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;im\_m \cdot \left(0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.02)
      (* (fma im_m (* im_m -0.16666666666666666) 1.0) (fma re im_m im_m))
      (if (<= t_0 0.0)
        (* im_m (* 0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))
        (*
         im_m
         (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(im_m, (im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m);
	} else if (t_0 <= 0.0) {
		tmp = im_m * (0.008333333333333333 * ((im_m * im_m) * (im_m * im_m)));
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	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(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(im_m, Float64(im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m));
	elseif (t_0 <= 0.0)
		tmp = Float64(im_m * Float64(0.008333333333333333 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.02], N[(N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(im$95$m * N[(0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 58.4

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

   5. Simplified 58.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. distribute-lft-out N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-lft-in N/A

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

   8. Simplified 17.3%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 48.5

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

   5. Simplified 48.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} \cdot {im}^{2} - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. *-commutative N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right) \]

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      11. *-lowering-*.f64 48.1

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

   8. Simplified 48.1%

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

   9. Taylor expanded in im around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{4}\right)} \]

       2. metadata-eval N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot {im}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \]

       3. pow-sqr N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \color{blue}{\left({im}^{2} \cdot {im}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot {im}^{2}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto im \cdot \left(\frac{1}{120} \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

       8. *-lowering-*.f64 27.0

          \[\leadsto im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \]

   11. Simplified 27.0%

       \[\leadsto im \cdot \color{blue}{\left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)} \]

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 54.1

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;im \cdot \left(0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* (fma im_m (* im_m -0.16666666666666666) 1.0) (fma re im_m im_m))
    (* im_m (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = fma(im_m, (im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m);
	} else {
		tmp = im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(fma(im_m, Float64(im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m));
	else
		tmp = Float64(im_m * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. distribute-lft-out N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-lft-in N/A

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

   8. Simplified 47.6%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 31.9

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \cdot im \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* (fma im_m (* im_m -0.16666666666666666) 1.0) (fma re im_m im_m))
    (* im_m (* re (fma re (fma re 0.16666666666666666 0.5) 1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = fma(im_m, (im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m);
	} else {
		tmp = im_m * (re * fma(re, fma(re, 0.16666666666666666, 0.5), 1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(fma(im_m, Float64(im_m * -0.16666666666666666), 1.0) * fma(re, im_m, im_m));
	else
		tmp = Float64(im_m * Float64(re * fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. distribute-lft-out N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

      14. *-rgt-identity N/A

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

      15. distribute-lft-in N/A

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

   8. Simplified 47.6%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 26.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, im\right), im\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), im\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot im + \left(\frac{1}{2} \cdot \frac{im}{re} + \frac{im}{{re}^{2}}\right)\right)} \]

   11. Simplified 32.5%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* im_m (fma im_m (* im_m -0.16666666666666666) 1.0))
    (* im_m (* re (fma re (fma re 0.16666666666666666 0.5) 1.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = im_m * fma(im_m, (im_m * -0.16666666666666666), 1.0);
	} else {
		tmp = im_m * (re * fma(re, fma(re, 0.16666666666666666, 0.5), 1.0));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im_m * Float64(re * fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 26.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, im\right), im\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), im\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot im + \left(\frac{1}{2} \cdot \frac{im}{re} + \frac{im}{{re}^{2}}\right)\right)} \]

   11. Simplified 32.5%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im\_m \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* im_m (fma im_m (* im_m -0.16666666666666666) 1.0))
    (* (* re re) (* im_m (fma re 0.16666666666666666 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 ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = im_m * fma(im_m, (im_m * -0.16666666666666666), 1.0);
	} else {
		tmp = (re * re) * (im_m * fma(re, 0.16666666666666666, 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 (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0));
	else
		tmp = Float64(Float64(re * re) * Float64(im_m * fma(re, 0.16666666666666666, 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[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 26.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, im\right), im\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), im\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot im + \frac{1}{2} \cdot \frac{im}{re}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + {re}^{3} \cdot \left(\frac{1}{2} \cdot \frac{im}{re}\right)} \]

       2. *-commutative N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{im}{re}\right) \cdot {re}^{3}} \]

       3. associate-*r/ N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \color{blue}{\frac{\frac{1}{2} \cdot im}{re}} \cdot {re}^{3} \]

       4. associate-*l/ N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \color{blue}{\frac{\left(\frac{1}{2} \cdot im\right) \cdot {re}^{3}}{re}} \]

       5. associate-/l* N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \color{blue}{\left(\frac{1}{2} \cdot im\right) \cdot \frac{{re}^{3}}{re}} \]

       6. unpow3 N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \frac{\color{blue}{\left(re \cdot re\right) \cdot re}}{re} \]

       7. unpow2 N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \frac{\color{blue}{{re}^{2}} \cdot re}{re} \]

       8. associate-*l/ N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{\left(\frac{{re}^{2}}{re} \cdot re\right)} \]

       9. *-rgt-identity N/A

          \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\frac{\color{blue}{{re}^{2} \cdot 1}}{re} \cdot re\right) \]

       10. associate-*r/ N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\color{blue}{\left({re}^{2} \cdot \frac{1}{re}\right)} \cdot re\right) \]

       11. unpow2 N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{re}\right) \cdot re\right) \]

       12. associate-*l* N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\color{blue}{\left(re \cdot \left(re \cdot \frac{1}{re}\right)\right)} \cdot re\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\left(re \cdot \color{blue}{1}\right) \cdot re\right) \]

       14. *-rgt-identity N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \left(\color{blue}{re} \cdot re\right) \]

       15. unpow2 N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{{re}^{2}} \]

       16. *-commutative N/A

           \[\leadsto {re}^{3} \cdot \left(\frac{1}{6} \cdot im\right) + \color{blue}{{re}^{2} \cdot \left(\frac{1}{2} \cdot im\right)} \]

   11. Simplified 27.8%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* im_m (fma im_m (* im_m -0.16666666666666666) 1.0))
    (* re (* im_m (* 0.16666666666666666 (* re 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 ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = im_m * fma(im_m, (im_m * -0.16666666666666666), 1.0);
	} else {
		tmp = re * (im_m * (0.16666666666666666 * (re * re)));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0));
	else
		tmp = Float64(re * Float64(im_m * Float64(0.16666666666666666 * Float64(re * 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[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. accelerator-lowering-fma.f64 26.0

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \color{blue}{\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)}, im\right), im\right) \]

   8. Simplified 26.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re \cdot im, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), im\right), im\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot im\right) \cdot {re}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{{re}^{3} \cdot \left(\frac{1}{6} \cdot im\right)} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(re \cdot re\right)\right)} \cdot \left(\frac{1}{6} \cdot im\right) \]

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

       11. unpow2 N/A

           \[\leadsto re \cdot \left(im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{6}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(re \cdot \left(re \cdot \frac{1}{6}\right)\right)}\right) \]

       13. *-commutative N/A

           \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot re\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto re \cdot \color{blue}{\left(im \cdot \left(re \cdot \left(\frac{1}{6} \cdot re\right)\right)\right)} \]

       15. *-commutative N/A

           \[\leadsto re \cdot \left(im \cdot \left(re \cdot \color{blue}{\left(re \cdot \frac{1}{6}\right)}\right)\right) \]

       16. associate-*r* N/A

           \[\leadsto re \cdot \left(im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \frac{1}{6}\right)}\right) \]

       17. unpow2 N/A

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

       18. *-lowering-*.f64 N/A

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

       19. unpow2 N/A

           \[\leadsto re \cdot \left(im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \frac{1}{6}\right)\right) \]

       20. *-lowering-*.f64 28.3

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

   11. Simplified 28.3%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.04:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.3% accurate, 0.9× speedup

Math

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

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (* im_m (fma im_m (* im_m -0.16666666666666666) 1.0))
    (* im_m (fma re (fma re 0.5 1.0) 1.0)))))

C

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im_m * fma(re, fma(re, 0.5, 1.0), 1.0));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 51.1

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

   5. Simplified 51.1%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 36.0

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

   8. Simplified 36.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 60.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 49.2

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

   8. Simplified 49.2%

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

4. Final simplification 41.0%

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

Alternative 17: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.04:\\ \;\;\;\;im\_m \cdot \mathsf{fma}\left(im\_m, im\_m \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.04)
    (* im_m (fma im_m (* im_m -0.16666666666666666) 1.0))
    (* im_m (* 0.5 (* re 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 ((exp(re) * sin(im_m)) <= 0.04) {
		tmp = im_m * fma(im_m, (im_m * -0.16666666666666666), 1.0);
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.04)
		tmp = Float64(im_m * fma(im_m, Float64(im_m * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im_m * Float64(0.5 * Float64(re * 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[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.04], N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0400000000000000008

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\sin im} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 59.5

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

   5. Simplified 59.5%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   if 0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 41.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 24.3

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

   8. Simplified 24.3%

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

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 24.7

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

   11. Simplified 24.7%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.04:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.45:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.45) im_m (* im_m (* 0.5 (* re 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 ((exp(re) * sin(im_m)) <= 0.45) {
		tmp = im_m;
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= 0.45d0) then
        tmp = im_m
    else
        tmp = im_m * (0.5d0 * (re * re))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= 0.45) {
		tmp = im_m;
	} else {
		tmp = im_m * (0.5 * (re * re));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= 0.45:
		tmp = im_m
	else:
		tmp = im_m * (0.5 * (re * re))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.45)
		tmp = im_m;
	else
		tmp = Float64(im_m * Float64(0.5 * Float64(re * re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= 0.45)
		tmp = im_m;
	else
		tmp = im_m * (0.5 * (re * re));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.45], im$95$m, N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.450000000000000011

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 74.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 41.4%

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

   if 0.450000000000000011 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 48.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 28.4

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

   8. Simplified 28.4%

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

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 28.7

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

   11. Simplified 28.7%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.45:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.45:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(re \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.45) im_m (* re (* 0.5 (* 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 ((exp(re) * sin(im_m)) <= 0.45) {
		tmp = im_m;
	} else {
		tmp = re * (0.5 * (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 ((exp(re) * sin(im_m)) <= 0.45d0) then
        tmp = im_m
    else
        tmp = re * (0.5d0 * (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.exp(re) * Math.sin(im_m)) <= 0.45) {
		tmp = im_m;
	} else {
		tmp = re * (0.5 * (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.exp(re) * math.sin(im_m)) <= 0.45:
		tmp = im_m
	else:
		tmp = re * (0.5 * (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(exp(re) * sin(im_m)) <= 0.45)
		tmp = im_m;
	else
		tmp = Float64(re * Float64(0.5 * 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 ((exp(re) * sin(im_m)) <= 0.45)
		tmp = im_m;
	else
		tmp = re * (0.5 * (re * im_m));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.45], im$95$m, N[(re * N[(0.5 * N[(re * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.450000000000000011

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 74.9%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 41.4%

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

   if 0.450000000000000011 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 48.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 28.4

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

   8. Simplified 28.4%

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

   9. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 21.7

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

   11. Simplified 21.7%

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

Alternative 20: 100.0% accurate, 1.0× speedup

Math

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

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 (* (exp re) (sin 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 * (exp(re) * sin(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 * (exp(re) * sin(im_m))
end function

Java

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

Python

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 21: 30.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 175:\\ \;\;\;\;im\_m\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\_m\\ \end{array} \end{array} \]

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 (<= im_m 175.0) im_m (* 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 (im_m <= 175.0) {
		tmp = im_m;
	} 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 (im_m <= 175.0d0) then
        tmp = im_m
    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 (im_m <= 175.0) {
		tmp = im_m;
	} 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 im_m <= 175.0:
		tmp = im_m
	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 (im_m <= 175.0)
		tmp = im_m;
	else
		tmp = 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 (im_m <= 175.0)
		tmp = im_m;
	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[im$95$m, 175.0], im$95$m, N[(re * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 175

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 79.2%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im} \]
   7. Step-by-step derivation

   8. Simplified 42.4%

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

   if 175 < im

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Simplified 35.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 7.4

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

   8. Simplified 7.4%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 8.1

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

   11. Simplified 8.1%

       \[\leadsto \color{blue}{re \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 30.1% accurate, 22.9× speedup

Math

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

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 (* im_m (+ re 1.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 58.3

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

5. Simplified 58.3%

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

6. Taylor expanded in im around 0

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

8. Simplified 35.3%

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

9. Final simplification 35.3%

   \[\leadsto im \cdot \left(re + 1\right) \]
10. Add Preprocessing

Alternative 23: 30.1% accurate, 29.4× speedup

Math

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

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 (fma re im_m 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 * fma(re, im_m, im_m);
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * fma(re, im_m, 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 + im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

5. Simplified 69.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 35.3

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

8. Simplified 35.3%

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

Alternative 24: 27.2% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 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 * 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 * im$95$m), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

5. Simplified 69.4%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{im} \]
7. Step-by-step derivation

8. Simplified 33.4%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))